Unit equation

A unit equation is an equation that expresses a relationship between physical units or units of measurement. For a known functional relationship, it contains the units of the physical quantities and physical constants occurring in the size equation . It is used to check the established function, to determine the unit of a variable or constant in a selected system of units or to convert between units.

Basics

Every special value of a physical quantity (size value) is specified as the product of a numerical value and a unit. For this purpose, a basic unit or a coherent derived unit can be specified for each quantity . In the international system of units, this is a product of all seven basic units of the system of units, each with its own power . If the unit of a size is marked with square brackets around the size symbol, this generally applies to every derived unit ${\displaystyle Q}$

${\displaystyle [Q]=\xi \cdot \mathrm {m} ^{\alpha }\cdot \mathrm {kg} ^{\beta }\cdot \mathrm {s} ^{\gamma }\cdot \mathrm {A} ^{\delta }\cdot \mathrm {K} ^{\epsilon }\cdot \mathrm {mol} ^{\zeta }\cdot \mathrm {cd} ^{\eta }}$

There is a number factor in it; for the coherent units dealt with almost exclusively here . The exponents to are whole numbers . In other systems of units, rational numbers can also appear as exponents. The exponents can be zero if the associated base units do not appear in the derived unit. ${\displaystyle \xi }$ ${\displaystyle \xi =1}$ ${\displaystyle \alpha }$ ${\displaystyle \eta }$

With this in mind, a unit equation is defined as a “mathematical relationship between base units, coherent derived units and other units of measurement”.

Note: According to the rules for formula sets, size symbols are written in italics (oblique) and unit symbols are written vertically (straight).

Unit consideration

A simple method of checking whether an equation with physical quantities can be correct at all is to set up the unit equation, and to check that the summands and the left-hand side of the equation match the right-hand side. If necessary, derived units must be replaced by their product of the base units in order to be able to simplify. However, this consideration is not sufficient to check a function as a whole for correctness; agreement is only a necessary, but not a sufficient condition .

The unit volt occurs particularly frequently in electrical engineering; it is common to think of this unit as a base unit and not to replace it. For the relationship between the "mechanical units" newton ( ), meter ( ), second ( ) and the "electrical units" watt ( ), volt ( ), ampere ( ), the exact definition applies ${\displaystyle \mathrm {N} }$ ${\displaystyle \mathrm {m} }$ ${\displaystyle \mathrm {s} }$ ${\displaystyle \mathrm {W} }$ ${\displaystyle \mathrm {V} }$ ${\displaystyle \mathrm {A} }$

${\displaystyle \mathrm {N\,m=W\,s=V\,A\,s} }$

Examples

Verification through unit consideration

Whom in the equation that supposedly applies to the ideal resonant circuit

${\displaystyle f_{0}={\frac {1}{2\pi {\sqrt {LC}}}}}$

it is questionable whether the factor is correct, a unit consideration is a simple test. With ${\displaystyle {\sqrt {LC}}}$

• the frequency and ${\displaystyle f_{0}}$ ${\displaystyle [f_{0}]=\mathrm {Hz=s^{-1}} }$
• the inductance and ${\displaystyle L}$ ${\displaystyle [L]=\mathrm {H={\frac {Vs}{A}}} }$
• the capacity and ${\displaystyle C}$ ${\displaystyle [C]=\mathrm {F={\frac {As}{V}}} }$

the unit equation and the following substitution result

${\displaystyle \mathrm {[f_{0}]={\sqrt {{\frac {1}{[L]}}\cdot {\frac {1}{[C]}}}}\quad \Rightarrow \quad s^{-1}={\sqrt {{\frac {A}{Vs}}\cdot {\frac {V}{As}}}}={\sqrt {\frac {1}{s^{2}}}}} }$

Since this calculation does not lead to a contradiction, nothing speaks against the correctness of the factor.

Determination of a unit

In order to determine the unit of a quantity, when setting up the defining function, all quantities that occur in it are initially reduced to the most fundamental quantities possible. Alternatively, all units of the occurring variables are rewritten to base units. Using the example of performance, this means: ${\displaystyle P}$

The achievement is the quotient of work and time , whereby the work is the product of force and distance . The force, in turn, is the product of mass and acceleration : ${\displaystyle W}$ ${\displaystyle t}$ ${\displaystyle F}$ ${\displaystyle s}$ ${\displaystyle m}$ ${\displaystyle a}$

${\displaystyle P={\frac {W}{t}}={\frac {F\,s}{t}}={\frac {m\,a\,s}{t}}}$

To determine the unit of the left side, the individual units of the right side are used

${\displaystyle [F]=\mathrm {N=kg\;{\frac {m}{s^{2}}}} }$,   ,   ,   ,   ${\displaystyle [s]=\mathrm {m} }$ ${\displaystyle [t]=\mathrm {s} }$ ${\displaystyle [m]=\mathrm {kg} }$ ${\displaystyle [a]=\mathrm {\frac {m}{s^{2}}} }$

inserted into the unit equation and summarized as much as possible:

${\displaystyle [P]={\frac {[F]\,[s]}{[t]}}={\frac {[m]\,[a]\,[s]}{[t]}}=\mathrm {{\frac {kg\;{\frac {m}{s^{2}}}\;m}{s}}={\frac {kg\cdot m^{2}}{s^{3}}}=kg^{1}\cdot m^{2}\cdot s^{-3}} }$

It is also known that the unit watt is also used for converted power . For this applies with the definition given above

${\displaystyle \mathrm {W={\frac {N\,m}{s}}={\frac {kg\,m^{2}}{s^{3}}}} }$

in accordance with the unit of calculated here . If this match had not been found, it would be an indication that the equation used to determine performance is incorrect. Conversely, the somewhat bulky unit calculated here can be replaced by the more convenient symbol of the derived unit watt. ${\displaystyle P}$

The unit one

Some variables are specified without a unit, for example the plane angle, the number of turns in a coil or quotients of similar variables such as efficiency. In these cases, in the general unit equation given above , all exponents are zero. The unit is thus the number one, unit symbol 1. It is usually not recorded. Depending on the circumstances, an auxiliary unit is used instead in order to nevertheless be able to identify a size value. For example, the unit radians is often used for the angle with or the unit micrometer per meter for the expansion, or the auxiliary unit of percent, which is useful for proportions of similar sizes, for the efficiency . ${\displaystyle 1\,\mathrm {rad} =1}$ ${\displaystyle 10^{6}\,\mathrm {\tfrac {\mu m}{m}} =1}$ ${\displaystyle 100\,\%=1}$

Transcendent functions such as , or are only defined for an independent variable that has the unit one. The dependent variable also has the unit one. ${\displaystyle y=\log(x)}$ ${\displaystyle y=\exp(x)}$ ${\displaystyle y=\sin(x)}$ ${\displaystyle x}$ ${\displaystyle y}$

Here, too, a unit consideration can be used to check the correctness of an equation. For example, when a capacitor is discharged through a resistor , the electrical voltage runs over time as a decaying exponential function with time in the exponent. Who is no longer clear whether the equation ${\displaystyle U}$ ${\displaystyle t}$

${\displaystyle U=U_{0}\cdot \mathrm {e} ^{-{\frac {t}{RC}}}\quad }$ or ${\displaystyle \quad U=U_{0}\cdot \mathrm {e} ^{-t\cdot {RC}}}$

reads, the units should check: a capacitance has the unit and the resistance has the unit , thus the unit has  . In order for the exponent to have the unit one, only the first equation can - if at all - be correct. ${\displaystyle C}$ ${\displaystyle \mathrm {\tfrac {As}{V}} }$ ${\displaystyle R}$ ${\displaystyle \mathrm {\tfrac {V}{A}} }$ ${\displaystyle RC}$ ${\displaystyle \mathrm {s} }$

Correspondingly, it can only be wrong because is. It becomes correct with a relationship: ${\displaystyle \quad \ln U=\ln U_{0}-{\frac {t}{RC}}\quad }$ ${\displaystyle [U]\neq 1}$ ${\displaystyle \quad \ln {\frac {U}{U_{0}}}=-{\frac {t}{RC}}\ .}$

Conversion between units

In addition to the relationship between the units of different sizes, the relationship between different units of the same size is also called the unit equation. The conversion exists both between coherent derived units and base units such as

Electric voltage	Unit volt	${\displaystyle \mathrm {V=kg\,m^{2}\,s^{-3}\,A^{-1}} }$
energy	Unit joule	${\displaystyle \mathrm {J=W\,s=N\,m=kg\,m^{2}\,s^{-2}} }$

as well as between other units and coherent derived units or base units. There are conversion factors unequal to one, which can be whole powers of ten or other numerical values ​​such as

length	${\displaystyle \mathrm {1\ mm=10^{-3}\ m} }$
pressure	${\displaystyle \mathrm {1\ bar=10^{5}\ Pa=10^{5}\ N\,m^{-2}=10^{5}\ kg\,m^{-1}\,s^{-2}} }$
angle	${\displaystyle \mathrm {1^{\circ }=(\pi /180)\ rad=\pi /180} }$
energy	${\displaystyle \mathrm {1\ kWh=3{,}6\cdot 10^{6}\ J} }$

Individual evidence

1. ↑ DIN 1313: 1998: sizes . Cape. 9.3
2. ↑ ^{a b} Johannes Fischer: sizes and units of electricity theory . Springer, 1961, pp. 11-17
3. ↑ ^{a b} DIN EN ISO 80000-1: 2013, sizes and units - general , chap. 3.23
4. ↑ Heribert Genreith: Macroeconomic Field Theory . Books on Demand, 2011, p. 39
5. ↑ Winfried Storhas: Applied bio process development . Wiley – VCH, 2018, p. 120
6. ↑ Hansjürgen Bausch, Horst Steffen: Electrical engineering. 5th edition. Teubner, 2004, p. 44
7. ^ Paul Dobrinski, Gunter Krakau, Anselm Vogel: Physics for engineers. 8th edition. Teubner, p. 278
8. ↑ DIN EN ISO 80000-1: 2013, sizes and units - general , chap. 3.8 and 3.10
9. ↑ Wolfgang Geller: Thermodynamics for mechanical engineers. 5th edition. Springer Vieweg, 2015, p. 4

Web links

• Le Système international d'unités . 9e édition, 2019 (the so-called "SI brochure", French and English). (accessed on June 10, 2020)
• Correct use of quantities, units and equations (accessed November 9, 2017)