Here, under the inspiration of Quinn,we describe the construction of a tabletop LEGOwatt balance capable of measuring gram-level masses with a much more modest relative standard uncertainty of 1%. For the instrument described here, the cost of parts totaled about $650, but a similar device can be built for significantly less. A large portion of the cost is in the data acquisition system used to transfer the data to a computer. A recommended parts list is provided in the Appendix . We encourage readers to use this manuscript as general guidance for constructing such a device and by no means as a definitive prescription. There are many ways to build a watt balance, and we consider here a concept to highlight general considerations that are most important for success.

Certain commercial equipment, instruments, and materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

Certain commercial equipment, instruments, and materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

6. Certain commercial equipment, instruments, and materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

6. Certain commercial equipment, instruments, and materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

The watt balance, first conceived by Dr. Bryan Kibble in 1975, is a mass metrology apparatus that balances the weight of an object against an electromagnetic force generated by a current-carrying coil immersed in a magnetic field. By design, the watt balance toggles between two measurement modes and indirectly compares electrical power and mechanical power, measured in units of watts (hence the term “watt balance”). It is essentially a force transducer that can be calibrated solely in terms of electrical, optical, and frequency measurements. A few watt balances around the world have demonstrated the capability of measuring 1 kg masses with a relative uncertainty of a few parts in 10

The quest for a redefined International System of Units (SI) has been a formidable global undertaking. If the effort concludes as expected, sometime in 2018 the seven base units (meter, kilogram, second, ampere, kelvin, mole, and candela) that have formed the foundation of our unit system for over half a century will be redefined via seven reference constants. In terms of mass metrology, the present standard, forged in 1879 and named the International Prototype Kilogram (IPK), is the only mass on Earth defined with zero uncertainty. In the redefined system, the base unit kilogram will be redefined via a fixed value of the Planck constant, finally severing its ties to the IPK. Different experimental approaches can be used to realizemass from the fixed value of. At the National Institute of Standards and Technology (NIST), we have chosen to pursue the watt balance to realize the kilogram in the US after the redefinition.

II. BASIC WATT BALANCE THEORY Section: Choose Top of page ABSTRACT I. INTRODUCTION II. BASIC WATT BALANCE TH... << III. LEGO WATT BALANCE ME... IV. ELECTRONICS AND DATA ... V. MEASUREMENT VI. SUMMARY CITING ARTICLES

Although we understand that the reader is eager to hear about the LEGO watt balance, we will first explain the physics underpinning the professional watt balance. Several national metrology institutes worldwide have constructed watt balances and are presently pursuing ultra-high-precision mass measurements. These watt balances can measure masses ranging from 500 g to 1 kg and obtain relative standard uncertainties as small as a few parts in 108, or about 1 ×106 times smaller than that of the LEGO watt balance.

1 L) is moved at a vertical speed v through a magnetic field (flux density B) so that a voltage V is induced. The induced voltage is related to the velocity through the flux integral BL V = B L v . (1) Similarly, force mode is also based on Lorentz forces. The gravitational force on a mass m is counteracted by an upward electromagnetic force F generated by the now-current-carrying coil in a magnetic field F = B L I = m g , (2) where g is the local gravitational field strength and I is the current in the coil. Even though a watt balance might appear functionally similar to an equal-arm balance, an equal-arm balance is passive, relying on comparing an unknown mass to a calibrated one, while a watt balance is active, relying on compensating the unknown weight with a known force. In this case, the weight of an object is compensated by a precisely adjusted electromagnetic force. The experiment involves two modes of operation: velocity mode and force mode (see Fig.). Velocity mode is based on the principle of Lorentz forces. A coil (wire length) is moved at a vertical speedthrough a magnetic field (flux density) so that a voltageis induced. The induced voltage is related to the velocity through the flux integralSimilarly, force mode is also based on Lorentz forces. The gravitational force on a massis counteracted by an upward electromagnetic forcegenerated by the now-current-carrying coil in a magnetic fieldwhereis the local gravitational field strength andis the current in the coil.

B and L can be measured accurately. Because both of these variables are difficult to measure precisely, velocity mode is necessary as a calibration technique. By combining Eqs. BL factor common to both equations, and rearranging the variables, expressions for electrical and mechanical power are equated and a solution for mass is obtained V I = m g v ⇒ m = V I g v . (3) In principle, mass could be realized solely by operating in force mode, but this requires thatandcan be measured accurately. Because both of these variables are difficult to measure precisely, velocity mode is necessary as a calibration technique. By combining Eqs. (1) and (2) , canceling out thefactor common to both equations, and rearranging the variables, expressions for electrical and mechanical power are equated and a solution for mass is obtained

The equation above relates mechanical power to electrical power and provides a means to relate mass to electrical quantities. The relationship equates “virtual” power, in the sense that the factors of each product, V and I or mg and v, are not measured simultaneously, but separately in the two modes. The “power” only exists virtually, i.e., as a mathematical product. The practical significance of a virtual comparison is that the result is independent of several friction terms such as the mechanical friction during velocity mode or the electrical resistance of the coil wire.

In order to make the connection from mass to the Planck constant through the electrical quantities, it is necessary to understand two quantum physical effects that have revolutionized electrical metrology since the second half of the last century: the Josephson effect and the quantum Hall effect. These two phenomena are what permit the measurement of electrical quantities in terms of the Planck constant to the precision required for the watt balance and redefinition. On a side note, another constant—the elementary charge e—is present in both the Josephson effect and the quantum Hall effect. However, in the final watt balance equation the elementary charge drops out.

f, a bias current is forced through this junction and a voltage of V = h 2 e f ≡ K J − 1 f (4) will develop across the junction. The quotient K J = 2e/h is named the Josephson constant in honor of Brian Josephson, who predicted this effect in 1960. 7 The Josephson effect and e/h ,” Am. J. Phys. 38, 1071– 1095 (1970). 7. J. Clarke, “,” Am. J. Phys., 1071–(1970). https://doi.org/10.1119/1.1976556 μV, so in order to build a practical voltage standard, tens of thousands of these junctions are connected in series on a single chip. At NIST, 8 A 10V programmable Josephson voltage standard and its applications for voltage metrology ,” Metrologia 49, 635– 643 (2012). 8. Y. Tang, V. N. Ojha, S. Schlamminger, A. Rüfenacht, C. Burroughs, P. Dresselhaus, and S. Benz, “,” Metrologia, 635–(2012). https://doi.org/10.1088/0026-1394/49/6/635 The Josephson effect can be observed in a Josephson voltage standard, which consists of two superconducting materials separated by a thin non-superconducting barrier. At superconducting temperatures, and while irradiating the junction with an electrical field at a microwave frequency, a bias current is forced through this junction and a voltage ofwill develop across the junction. The quotient= 2is named the Josephson constant in honor of Brian Josephson, who predicted this effect in 1960.One junction delivers only a small voltage, typically 37V, so in order to build a practical voltage standard, tens of thousands of these junctions are connected in series on a single chip. At NIST,a chip the size of half a business card with approximately 250,000 junctions is immersed in liquid helium and can produce any voltage up to 10 V with an uncertainty of 1 nV. In principle, the Josephson voltage standard is a digitally adjustable battery—with a ∼$100,000 price tag.

V H occurs perpendicular to the magnetic flux and the current. While in the classical Hall effect the conductor immersed is a three-dimensional object, in the quantum Hall effect, the electrical conduction is confined to two dimensions. In such a system and at sufficiently high magnetic field, the ratio between the Hall voltage and current—the Hall resistance R H —becomes quantized to R H = V H I = 1 i h e 2 ≡ 1 i R K , (5) where i is an integer. The quotient R K = h/e2 is named the von Klitzing constant to honor Klaus von Klitzing, who first discovered this effect in 1980 (see Ref. 9 The quantum Hall effect ,” Am. J. Phys. 61, 179– 183 (1993). 9. J. Eisenstein, “,” Am. J. Phys., 179–(1993). https://doi.org/10.1119/1.17280 10 The ampere and electrical standards ,” J. Res. Nat. Inst. Stand. Technol. 106, 65– 103 (2001). 10. R. E. Elmquist, M. E. Cage, Y.-H. Tang, A.-M. Jeffery, J. R. J. Kinard, R. F. Dziuba, N. M. Dziuba, and E. R. Williams, “,” J. Res. Nat. Inst. Stand. Technol., 65–(2001). https://doi.org/10.6028/jres.106.005 9. On the outside, a quantum Hall system looks similar to a Josephson voltage system: a bundle of cables leading into a liquid helium dewar. On the inside, a fingernail-sized chip sits in a strong magnetic field at temperatures below 1.5 K. A skilled operator can use the device to realize the same resistance value independent of time and place. The quantum Hall effect is a special case of the Hall effect. The Hall effect occurs when a current-carrying conductor is immersed in a magnetic field and a Hall voltageoccurs perpendicular to the magnetic flux and the current. While in the classical Hall effect the conductor immersed is a three-dimensional object, in the quantum Hall effect, the electrical conduction is confined to two dimensions. In such a system and at sufficiently high magnetic field, the ratio between the Hall voltage and current—the Hall resistance—becomes quantized towhereis an integer. The quotientis named the von Klitzing constant to honor Klaus von Klitzing, who first discovered this effect in 1980 (see Ref.). At NIST, the quantum Hall effect is the starting point of resistance dissemination.Scaling with a cryogenic current comparator allows researchers to measure a 100 Ω precision resistor with a relative uncertainty of a few parts in 10. On the outside, a quantum Hall system looks similar to a Josephson voltage system: a bundle of cables leading into a liquid helium dewar. On the inside, a fingernail-sized chip sits in a strong magnetic field at temperatures below 1.5 K. A skilled operator can use the device to realize the same resistance value independent of time and place.

6 times smaller than that of the LEGO watt balance built at home or in the classroom. You may be wondering why all of a sudden we need to make a resistance (R) measurement when we actually need a current (I) measurement. Because a high-precision measurement of I is difficult to achieve, we simply use Ohm's law and equate I = V/R. Hence, instead of measuring P = VI, the current I is driven through a precisely calibrated resistor R, producing a voltage drop V R , yielding P = VV R /R. Both voltages are measured by comparing to a Josephson voltage standard, so their values can be expressed in terms of a frequency and the Josephson constant. The resistor is measured by comparing to a quantum Hall resistor, so its value can be expressed in terms of R K . This can be written as P = V V R / R = C f 1 f 2 h 2 e h 2 e e 2 h = C f 1 f 2 4 h . (6) Here, C is a known constant that indicates the number of junctions used and the ratio of R to h/e2. Combining the above equation with Eq. h = 4 C f 1 f 2 m g v ⇒ m = C f 1 f 2 4 h g v . (7) Before the 2018 redefinition of units, the equation on the left is used to measure h from a mass traceable to the IPK. After redefinition, the equation on the right will be used to realize the definition of the kilogram from a fixed value of h in joule-seconds. Together, these two quantum electrical standards enable scientists at NIST to build a watt balance with a relative measurement uncertainty that is about 1 × 10times smaller than that of the LEGO watt balance built at home or in the classroom. You may be wondering why all of a sudden we need to make a resistance () measurement when we actually need a current () measurement. Because a high-precision measurement ofis difficult to achieve, we simply use Ohm's law and equate. Hence, instead of measuring, the currentis driven through a precisely calibrated resistor, producing a voltage drop, yielding. Both voltages are measured by comparing to a Josephson voltage standard, so their values can be expressed in terms of a frequency and the Josephson constant. The resistor is measured by comparing to a quantum Hall resistor, so its value can be expressed in terms of. This can be written asHere,is a known constant that indicates the number of junctions used and the ratio ofto. Combining the above equation with Eq. (3) yieldsBefore the 2018 redefinition of units, the equation on the left is used to measurefrom a mass traceable to the IPK. After redefinition, the equation on the right will be used to realize the definition of the kilogram from a fixed value ofin joule-seconds.

11,12 News from the BIPM ,” Metrologia 26, 69– 74 (1989). 11. T. Quinn, “,” Metrologia, 69–(1989). https://doi.org/10.1088/0026-1394/26/1/006 New international electrical reference standards based on the Josephson and quantum Hall effects ,” Metrologia 26, 47– 62 (1989). 12. B. Taylor and T. Witt, “,” Metrologia, 47–(1989). https://doi.org/10.1088/0026-1394/26/1/004 K J–90 and R K–90 , respectively. Since 1990, almost all electrical measurements are calibrated in conventional units. By comparing electrical power in conventional units to mechanical power in SI units, h can be determined. In a classroom setting, quantum electrical standards are typically unavailable. However, it is still possible to measure the Planck constant due to the way the present unit system is structured. While the SI is used for most measurements, a different system of units has been used worldwide for almost all electrical measurements since 1990. For these so-called conventional units, the Josephson and von Klitzing constants were fixed at values adjusted to the best knowledge in 1989.These fixed values are named “conventional Josephson” and “conventional von Klitzing” constants and are abbreviatedand, respectively. Since 1990, almost all electrical measurements are calibrated in conventional units. By comparing electrical power in conventional units to mechanical power in SI units,can be determined.

V I = m g v ⇒ { V I } 90 W 90 = { m g v } SI W SI , (8) where {x} 90 and {x} SI denote the numerical values of the quantity x in conventional and SI units, respectively. Further, W 90 and W SI are the units of power (watt) in the conventional and SI systems. The equation above can be written as { m g v } SI { V I } 90 = W 90 W SI = h h 90 ⇒ h = h 90 { m g v } SI { V I } 90 , (9) where h 90 is the conventional Planck constant, defined as h 90 ≡ 4 K J – 90 2 R K – 90 = 6.626 068 854 … × 10 − 34 J s . (10) Thus, the value of the Planck constant can be determined by multiplying the conventional Planck constant by the ratio of mechanical power in SI units to electrical power in conventional units. Starting at Eq. (3) , we see thatwhere {and {denote the numerical values of the quantityin conventional and SI units, respectively. Further, Wand Ware the units of power (watt) in the conventional and SI systems. The equation above can be written aswhereis the conventional Planck constant, defined asThus, the value of the Planck constant can be determined by multiplying the conventional Planck constant by the ratio of mechanical power in SI units to electrical power in conventional units.

BL to each mode, i.e., ( B L ) v = V v and ( B L ) F = m g I . (11) Using these two numbers, the ratio of h/h 90 is given by h h 90 = ( B L ) F ( B L ) v = { m g v } SI { V I } 90 . (12) To arrive at this ratio, we start by assigning different flux integralsto each mode, i.e.,Using these two numbers, the ratio ofis given by

m = V I g v , (13) where all quantities are expressed in SI units. After redefinition, electrical power and mechanical power will be measured in the same units and the schism between units will vanish. Then, referring back to Eq. (3) , an arbitrary mass can be determined using a watt balance simply aswhere all quantities are expressed in SI units.

g and v are measured accurately by NIST scientists with an absolute gravimeter and interferometric methods, respectively. However, since this manuscript's main focus is still a proof-of-principle LEGO watt balance, ultra-high-precision metrology approaches are unnecessary. Gravity can be estimated by inputting one's geographical coordinates into the web page found in Ref. 13 Surface gravity prediction by NGS software requests ,” < 13. National Oceanic and Atmospheric Administration, “,” < https://www.ngs.noaa.gov/cgi-bin/grav_pdx.prl >. The remaining two variablesandare measured accurately by NIST scientists with an absolute gravimeter and interferometric methods, respectively. However, since this manuscript's main focus is still a proof-of-principle LEGO watt balance, ultra-high-precision metrology approaches are unnecessary. Gravity can be estimated by inputting one's geographical coordinates into the web page found in Ref., or even measured experimentally with a simple pendulum in the laboratory. Velocity can be determined using a simple optical method that we describe in Sec. V

However, do not be fooled by our toy. The LEGO watt balance is versatile and fully capable of measuring in either mode. It will be a device to measure the Planck constant before redefinition and one to realize mass after redefinition. A capable operator can perform a measurement with a relative uncertainty of 1% with the device described below.