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RADIO & ELECTRONICS FORMULAS & DATA
Radio & Electronics Formulas & Data
- a quick reference for radio and electronics, data, formulae or formulas for calculations
that are so easily forgotten.
Resistors, resistance, capacitors, & capacitance formulas and equations
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Resistivity basics
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Resistivity table / chart
Resistance temperature coefficient
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Electrical Resistivity Basics
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- details of resistivity including resistivity formula / equation and resistivity units used.
Materials naturally have some resistance to the flow of an electric current.
Some materials resist the flow of current more than others, and therefore it is known as the electrical resistivity of the substance.
The term electrical resistivity may also be known as the specific electrical resistance, or volume resistivity in some circumstances.
In addition to this, specific units are used for resistivity, and there are formulae or equations for calculating resistivity.
Resistivity basics
The electrical resistivity of a material is also known as its specific electrical resistance. It is a measure of how strongly a material opposes the flow of
electric current.
A definition of resistivity is the electrical resistance per unit length and per unit of cross-sectional area. This is for a particular material at a specified
temperature.
It is also possible to define the resistivity of a substance as the resistance of a cube of that substance having edges of unit length, with the understanding
that the current flows normal to opposite faces and is distributed uniformly over them. The SI unit for electrical resistivity is the ohm metre, although it is
also sometimes specified in ohm centimetres.
This means that a low resistivity indicates a material that readily allows the movement of electrons. Conversely a high resistivity material will have a high
electrical resistance and will impede the flow of electrons.
Elements such as copper and aluminium are known for their low levels of resistivity. Silver and in particular, gold have a very low resistivity, but for
obvious cost reasons their use is restricted.
Resistivity units
The SI unit of electrical resistivity is the ohm⋅metre (Ω⋅m). It is commonly represented by the Greek letter ρ, rho.
Although the SI resistivity unit, the ohms metre is generally used, sometimes figures will be seen described in terms of ohms centimetres, Ω⋅cm.
Resistivity equation / formula
The resistivity of a material is defined in terms of the magnitude of the electric field across it that gives a certain current density.
Where:
ρ is the resistivity of the material in ohm metres, Ω⋅m
E is the magnitude of the electric field in volts per metre, V⋅m^-1
J is the magnitude of the current density in amperes per square metre, A⋅m^-2
Many resistors and conductors have a uniform cross section with a uniform flow of electric current. It is therefore possible to create the more specific, but
more widely used equation:
Where:
R is the electrical resistance of a uniform specimen of the material measured in ohms
l is the length of the piece of material measured in metres, m
A is the cross-sectional area of the specimen measured in square metres, m^2
Material resistivity levels
Materials are put into different categories according to their level or resistivity. A summary is given in the table below.
RESISTIVITY REGIONS FOR DIFFERENT CATEGORIES OF MATERIALS
MATERIAL TYPE
RESISTIVITY REGION
Electrolytes
Variable*
Insulators
~10^16
Metals
~10^-8
Semiconductors
Variable*
Superconductors
0
*The level of conductivity of semiconductors is dependent upon the level of doping. With no doping they appear almost like an insulator, but with doping
charge carriers are available and the resistivity falls dramatically. Similarly for electrolytes, the level of resistivity varies widely.
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RADIO & ELECTRONICS FORMULAS & DATA
Resistivity table / chart
- table or chart of the resistivity of a variety of materials many of which are used in
electronics equipment.
The resistivity table given below contains many of the substances widely used in electronics.
These two properties are particularly important and will often determine whether a substance can be used in the manufacture of a wide variety of electrical
and electronics components from wire to components such as resistors, potentiometers and many more.
TABLE OF RESISTIVITY
FOR DIFFERENT SUBSTANCES
SUBSTANCE
RESISTANCE AT 20°C
OHM METRES
Aluminium
2.8 x 10^-8
Antimony
3.9 x 10^-7
Bismuth
1.3 x 10^-6
Brass
Cadmium
~0.6 - 0.9 x 10^-7
6 x 10^-8
Cobalt
5.6 x 10^-8
Copper
1.7 x 10^-8
Gold
2.4 x 10^-8
Carbon (Graphite)
Germanium
1 x 10^-5
4.6 x 10^-1
TABLE OF RESISTIVITY
FOR DIFFERENT SUBSTANCES
SUBSTANCE
RESISTANCE AT 20°C
OHM METRES
Iron
1.0 x 10^-7
Lead
1.9 x 10^-7
Manganin
4.2 x 10^-7
Nichrome
1.1 x 10^-6
Nickel
7 x 10^-8
Palladium
1.0 x 10^-7
Platinum
0.98 x 10^-7
Quartz
7 x 10^17
Silicon
6.4 x 10^2
Silver
1.6 x 10^-8
Tantalum
1.3 x 10^-7
Tin
1.1 x 10^-7
Tungsten
4.9 x 10^-8
Zinc
5.5 x 10^-8
Many of the materials found in the resistivity table above are widely used in electronics. Aluminium and particularly copper are used for their low levels of
resistance. Most wire used these days for interconnections is made from copper as it offers a low level of resistivity at an acceptable cost. Gold while much
better is more costly and is used in much smaller quantities. Often gold plating is found on high quality low current connectors where it ensures the lowest
contact resistance. Silver is not so widely used because it tarnishes and this can result in higher contact resistances. The oxide can also under some
circumstances act as a rectifier which may cause some annoying problems in RF circuits.
Tantalum also appears in the resistivity table. It is used in capacitors, and nickel and palladium are used in the end connections for many surface mount
components such as capacitors. Quartz finds its main use as a piezo-electric resonant element. Quartz crystals are sued as frequency determining elements
in many oscillators where its high value of Q enables very frequency stable circuits to be made. They are similarly used in high performance filters.
Temperature Coefficient of Resistance
- formula as well as table of values for the temperature coefficient of resistance for
various substances - the resistance temperature dependence.
The resistance of all substances varies with temperature. This temperature resistance dependence has a bearing on electronic circuits in many ways.
In most cases the resistance increases with temperature, but in some it falls.
As a result it is often necessary to have an understanding of the resistance temperature dependence.
Temperature coefficient of resistance basics
The reason behind the temperature coefficient of resistance within a conductor can be reasoned intuitively.
The resistance of a material has a dependence upon a number of phenomena. One of these is the number of collisions that occur between the charge carriers
and atoms in the material. As the temperature increases so do the number of collisions and therefore it can be imagined that there will be a marginal
increase in resistance with temperature.
This may not always be the case because some materials have a negative temperature coefficient of resistance. This can be caused by the fact that with
increasing temperature further charge carriers are released which will result in a decrease in resistance with temperature. As might be expected, this effect
is often seen in semiconductor materials.
When looking at the resistance temperature dependence, it is normally assumed that the temperature coefficient of resistance follows a linear law. This is
the case around room temperature and for metals and many other materials. However it has been discovered that the resistance effects resulting from the
number of collisions is not always constant, particularly at very low temperatures for these materials.
The resistivity has been shown to be inversely proportional to the mean free path between collisions, i.e. this results in increasing resistivity / resistance
with increasing temperature. For temperatures above about 15°K (i.e. above absolute zero), this is limited by thermal vibrations of the atoms and this gives
the linear region which we are familiar. Below this temperature, the resistivity is limited by impurities and available carriers.
Resistance temperature graph
Temperature coefficient of resistance formula
The resistance of a conductor at any given temperature can be calculated from a knowledge of the temperature, its temperature coefficient of resistance, its
resistance at a standard temperature, and the temperature of operation. The equation for this resistance temperature dependence can be expressed in general
terms as:
Where
R = the resistance at temperature, T
Rref = the resistance at temperature Tref
α = the temperature coefficient of resistance for the material
T = the material temperature in ° Celcius
Tref = is the reference temperature for which the temperature coefficient is specified.
The temperature coefficient of resistance is normally standardised in relation to a temperature of 20°C as this is normal "room temperature." Accordingly
the equation normally used in practical terms is:
Where
R20 = the resistance at 20°C
α20 is the temperature coefficient of resistance at 20°C
Temperature coefficient of resistance table
The table below gives the temperature coefficient of resistance for a variety of substances including the copper temperature coefficient of resistance, etc..
TEMPERATURE COEFFICIENT OF RESISTANCE TABLE
FOR DIFFERENT SUBSTANCES
SUBSTANCE
TEMPERATURE COEFFICIENT
°C^-1
Aluminium
43 x 10^-4
(18°C - 100°C)
Antimony
40 x 10-4
Bismuth
42 x 10-4
Brass
~10 x 10^-4
Cadmium
40 x 10-4
Cobalt
7 x 10-5
Constantan (Alloy)
33 x 10-4
TEMPERATURE COEFFICIENT OF RESISTANCE TABLE
FOR DIFFERENT SUBSTANCES
SUBSTANCE
TEMPERATURE COEFFICIENT
°C^-1
Copper
40 x 10^-4
Gold
34 x 10^-4
Carbon (Graphite)
-5.6 x 10^-4
Germanium
-4.8 x 10^-2
Iron
56 x 10^-4
Lead
39 x 10^-4
Manganin
~2 x 10^-5
Molybdenum
46 x 10^-4
Nichrome
1.7 x 10^-4
Nickel
59 x 10^-4
Platinum
38 x 10^-4
Silicon
-7.5 x 10^24
Silver
40 x 10^-4
Tantalum
33 x 10-4
Tin
45 x 10^-4
Tungsten
45 x 10^-4
Zinc
36 x 10-4
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Electrical Conductivity Basics
- details of conductivity including conductivity formula / equation and conductance or
conductivity units used in terms of siemens or mhos.
Electrical conductivity or electrical conductance has a measure of how an electrical current moves within a substance. The higher the conductivity, the
greater the current density for a given applied potential difference.
The electrical conductivity or electrical conductance of a substance is a measure of the its ability to conduct electricity.
The conductivity is important because some substances are required to conduct electricity as well as possible, i.e. in the case of wire conductors, whereas
others are used as insulators, and other substances may be required to conduct less electricity, acting as a resistor.
Electrical conductivity basics
Electrical conductivity is a ratio of the current density to the electric field strength. The higher the value of the conductivity, the lower the resistance it
provides to the flow of electric current.
The value of the electrical conductivity depends on the ability for electrons or other charge carriers such as holes to move within the lattice of the material.
Highly conductive materials such as copper allow the free movement of electrons within their molecular lattice. There are free electrons within the lattice.
Materials with a low level of conductivity or conductance have very few free electrons within their structure. Electrons are tightly held within the
molecular structure and require a significant level of energy to pull them free.
Electrical conductivity units
The electrical conductivity units are siemens per metre, S⋅m^-1.
The siemens also used to be referred to as a mho - this is the reciprocal of a an ohm, and this is inferred by spelling ohm backwards.
Conductance is the reciprocal of resistance and one siemens is equal to the reciprocal of one ohm, and is sometimes referred to as the mho.
The siemens was adopted by the 14th General Conference on Weights and Measures as an SI derived unit in 1971 - named after Ernst Werner von
Siemens.
As with every SI, International System of Units name that is derived from the proper name of a person, the first letter of its symbol is upper case, i.e. in this
case "S". When an SI unit is spelled out in English, it should always begin with a lower case letter, i.e. in this case Siemens. The exception to this is where
any word would be capitalised, as in the case of the beginning of a sentence, etc.
The symbol that is most commonly used is the lower case version of the Greek letter sigma, σ, although but kappa, &kappa, gamma, &gamma, are also
used on occasions.
Although the SI units for conductivity are most widely used, conductivity values are often stated in terms of their IACS percentage value. The IACS,
International Annealed Copper Standard, was established by the 1913 International Electrochemical Commission.
The conductivity of the annealed copper (5.8001 x 107S/m) is defined to be 100% IACS at 20°C.
All other conductivity values are related back to this conductivity value. This means that iron with a conductivity value of 1.04 x 107 S/m, has a
conductivity of approximately 18% of that of annealed copper and this is given as 18% IACS.
As metallic processing methods have improved since the introduction of the standard, some modern copper products now often have IACS conductivity
values greater than 100% IACS because more impurities can now be removed from the metal.
Electrical conductivity formulas and equations
Resistivity and conductivity are interrelated. Conductivity is the inverse of resistivity. Accordingly it is easy to express one in terms of the other.
Where:
σ is the conductivity of the material in siemens per metre, S⋅m^-1
ρ is the resistivity of the material in ohm metres, Ω⋅m
This identity can then be substituted into the equation of resistivity, and this reveals the following relationship.
Where:
σ is the conductivity of the material in siemens per metre, S⋅m^-1
E is the magnitude of the electric field in volts per metre, V⋅m^-1
J is the magnitude of the current density in amperes per square metre, A⋅m^-2
Often it is necessary to relate the conductivity to a specific length of material with a constant cross sectional area..
Using this diagram, it is possible to relate the conductivity to the resistance, length and cross sectional area of the specimen.
Where:
R is the electrical resistance of a uniform specimen of the material measured in ohms
l is the length of the piece of material measured in metres, m
A is the cross-sectional area of the specimen measured in square metres, m^2
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