Siemens
SI derived unit · quantity: electrical conductance
01 · Definition
One siemens is the conductance of a stretch whose resistance is one ohm. There is no new physics in the unit, there is convenience of writing: parallel conductances simply add up, whereas resistances have to be worked out through reciprocals. That is why people speak in siemens wherever everything is connected in parallel — in electrochemistry about solutions, in physiology about membranes, in electronics about the transconductance of a transistor.
That is exactly what justifies the unit. In a parallel connection resistances have to be combined through reciprocals, while conductances simply add up, like currents at a node; the calculation of a complex network in siemens comes down to addition. The second field of use is solutions. The electrical conductivity of water is proportional to the amount of dissolved salts, and all water treatment, from an aquarium to a nuclear plant, measures it in microsiemens per centimetre. In pure water the conductivity is set by its own dissociation and lies at about five hundredths of a microsiemens per centimetre — a limit no distillation goes below.
Three resistors of three ohms each, connected in parallel, give one ohm — but getting there through resistances means adding three fractions and turning the result over. In siemens the same thing is done in your head: a third plus a third plus a third makes one. On this rests the nodal method of circuit analysis, where the unknowns are the node potentials and the coefficients are the conductances of the branches: the matrix comes out symmetric and is solved numerically without any fractions at all. Power-system calculations are carried out in siemens, and converted to ohms only for the final answer.
02 · Conversion
Enter a conductance and the passport will turn it into resistance
The link with the ohm is reciprocal, so the table divides the unit by the value. The mho and the “reciprocal ohm” are the same siemens under old names, found in handbooks up to the nineteen-seventies.
A conductivity meter measures not the conductivity of the solution but the conductance of its own cell, so the sensor’s data sheet states a cell constant: the ratio of the distance between the electrodes to their area. Multiplying the measured siemens by this constant in reciprocal centimetres gives microsiemens per centimetre. For pure water cells with a small constant are used, for seawater ones with a large constant, otherwise the instrument runs off scale.
| Unit | Name | Value | Where it is met |
|---|---|---|---|
| S | siemens, the SI unit | 1 | the unit of this sheet |
| mS | millisiemens | 1000 | the settled stretch in electrochemistry |
| µS | microsiemens | 1000000 | per centimetre — the working notation of laboratories |
| nS | nanosiemens | 10⁹ | here conductance is already a nuisance rather than useful |
| kS | kilosiemens | 0.001 | resistance drops into microohms |
| Ω | ohm, resistance | 1 | a reciprocal quantity, not another unit |
| kΩ | kilohm | 0.001 | four decimal places — that is microsiemens already |
| ℧ | mho, ohm backwards | 1 | the same thing under an old name |
| abS | absiemens, CGSM | 10⁻⁹ | a billion siemens in a single unit |
| statS | CGSE siemens | 8.988·10¹¹ | a little more than a picosiemens |
| Siem. unit | Siemens unit of 1860 | 0.9536 | a mercury column one metre long |
| 2e²/h | conductance quantum | 12906 | an exact number since 2019 |
| S | siemens | 1 | the unit of this sheet |
The link with the ohm is reciprocal, so the table divides the unit by the value: parallel conductances add up, resistances do not
03 · Orders of magnitude
from insulator to superconductorOne siemens: a stretch with a resistance of one ohm
04 · Measuring instruments
What conductance is measured with
It passes alternating current through the solution and works out the conductance by Ohm’s law. Direct current will not do: it decomposes the water and coats the electrodes with the products of electrolysis, so the supply is always alternating, usually at a kilohertz.
The solution passes through two toroids: the first induces a ring current in the liquid, the second picks it up. The electrodes never touch the medium at all, so the sensor works in acids, alkalis and dirty effluent, where ordinary plates would be destroyed within a month.
Since 1978 the salinity of the ocean has been determined not by chemical analysis but by measuring conductivity together with temperature and pressure. The practical salinity scale is defined through the ratio of the sample’s conductivity to that of a reference potassium chloride solution, and is therefore dimensionless.
The conductance of a bridge one atom thick changes not smoothly but in steps that are multiples of the quantum, 77.48 microsiemens. The value depends only on the Planck constant and the charge of the electron, and the effect itself is watched by stretching a gold wire until it breaks.
05 · Writing rules
A capital S — otherwise it is the second
The case carries the whole load here: S is the siemens, s is the second. The name of the unit does not change in the plural in Russian — “five siemens”, not “siemenses”, because it is a surname. Until 1971 the same quantity was called the “mho” — ohm read backwards, with an inverted symbol ℧. The name held for almost a century and still turns up in English-language transistor documentation, where transconductance is given in millimhos. The conductivity of a solution is written in S/m, but laboratory practice holds firmly to µS/cm: it is a hundred times smaller and the figures come out handier.
The Russian symbol is См, with a capital С. Until 1971 the unit was called the “mho” — ohm read backwards — and written with an inverted omega ℧; the notation has been officially abolished but is met in handbooks on vacuum tubes and in English-language transistor literature, where transconductance is still sometimes given in millimhos. The conductivity of solutions is customarily expressed in microsiemens per centimetre rather than per metre: the factor between the two notations is a hundred, and confusing it gives an error that can easily pass boiler water as fit when it is not.
06 · Neighbouring units
The siemens and the ohm are one quantity from two sides. In alternating-current circuits the quantity reciprocal to impedance is called admittance and is also measured in siemens.
The conductance quantum is the reciprocal of half the von Klitzing constant: the two numbers describe one and the same quantum limit from opposite sides. From the Simetrium passports next to the siemens stand ohm as the reciprocal quantity, ampere s the volt from the definition, moles per litre for solutions, and farad from the reactive part of admittance.
07 · Historical section
The ohm read backwards
Siemens built the telegraph line from St Petersburg to the Crimea and the first electric tram. The unit of conductance is named after him — and it is the reciprocal of the ohm.
Conductance used to be measured in “mhos” — ohm backwards — and written with an inverted omega. The name is credited to William Thomson, and it held for more than eighty years: valve handbooks are full of millimhos.
The Fourteenth General Conference adopted the siemens, the katal and the mole, closing the list of derived units with special names. The siemens turned out to be the only SI unit named after a businessman rather than a scientist alone.
In a point contact between two conductors the conductance changes in jumps that are multiples of the quantum, 77.48 microsiemens. Each step means one more channel opening for the electrons — the limit beyond which conductance stops being a continuous quantity.
One name went to two different units a hundred years apart
In 1860 Werner Siemens proposed a standard of resistance — a column of mercury one metre long with a cross-section of one square millimetre — and it was named the “Siemens unit”. It equalled 0.9536 of today’s ohm and for a quarter of a century served as the working measure of telegraph engineers across continental Europe. Twenty years later the congress chose the ohm instead, slightly different in size. And in 1971 the Fourteenth General Conference gave the name of Siemens to the unit of conductance — the quantity reciprocal to the one he had written about. The oddity is that a businessman proposed a unit of resistance, and his name went to its inverse.
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SI derived units with names of their own
the siemens passport is openSI base units
in terracotta — the four units the siemens is built fromConductance in other notations
Passport language
20 languages. The symbol S is international, but water hardness is not: German degrees, French, English and milligrams per litre — the passport brings them to local practice.