Resistors, resistance, capacitors, & capacitance formulas and equations

Electrical Resistivity Basics

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.

Resistivity table / chart

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	~0.6 - 0.9 x 10^-7
Cadmium	6 x 10^-8
Cobalt	5.6 x 10^-8
Copper	1.7 x 10^-8
Gold	2.4 x 10^-8
Carbon (Graphite)	1 x 10^-5
Germanium	4.6 x 10^-1
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

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
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

Electrical Conductivity Basics

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

Series and parallel resistors

Resistor networks consisting of resistors in series or resistors in parallel are often found in electronic circuits.

There are many reasons why parallel resistor networks or series resistor networks may be used. It is possible that the required value may not be available, and a combination of two or more may provide the required value. Alternatively using more than one resistor may enable the required overall wattage to be achieved.

These scenarios are often found when manufacturers need to limit the number of different types or values of resistor in a design as this helps reduce costs. In this way, using series or parallel resistor networks may provide significant advantages.

Resistors in series

Resistors can be placed in series or parallel. When placed in series the total resistance is equal to the sum of the individual resistors:

Resistors in series

It is also worth noting that the same current flows through each resistor, but the voltage across each resistor is proportional to the resistance of that particular resistor.

Resistors in parallel

For resistors placed in parallel, the arithmetic is a little more complicated because the reciprocal of the total resistance is equal to the sum of the reciprocals of the constituent resistors:

Resistors in parallel

As the voltage across all the resistors is the same and the current is shared according to the resistance of the individual resistors, the formula for calculating the overall resistance of the resistors in parallel is more complicated than the series resistor case and becomes:

Two resistors in parallel

In most cases there are only two resistors in a parallel network. Normally it is not necessary to involve more than two resistors.

In this case where there are only two resistors R1 and R2 in parallel the calculation can be simplified.

If both these resistors have the same value it can be seen that the overall value of the resistance is half the value for the individual resistor.

When resistors are placed in parallel the voltage across the resistors is the same, but the current through each one is inversely proportional to its resistance.

Parallel Resistor Values Table

While the values for resistors in parallel can be calculated relatively easily in most cases a table giving suitable values that are in the most case preferred is given below./p>

The table below gives a enables preferred values to be created from other preferred resistor values.

This can be very useful when trying to obtain an exact resistor value when the required ones are not available.

Parallel resistor values table

The table below gives values for resistors in parallel for a variety of values in the various tolerance, E-series categories.

Column A gives the required resistor value, and column B gives the first choice combination for the parallel resistors. A second choice is shown in column C. The values should be a close tolerance as possible, typically 0.5%.

The table below provides resistor values using standard values, even if they are in a higher tolerance series, E series than might otherwise be used.

PARALLEL RESISTOR VALUES TABLE
REQUIRED RESISTOR VALUE "A"	FIRST SELECTION OF RESISTORS "B"		ALTERNATIVE SELECTION OF RESISTORS "C"
10.0	10		11	110
11.0	11		12	130
12.0	12		13	150
13.0	13		15	100
14.0	16	110	15	220
15.0	15		16	240
16.0	16		27	39
17.0	22	75	18	300
18.0	18		20	180
19.0	24	91	20	390
20.0	20		22	220
22.0	22		24	270
23.0	39	56	24	560
24.0	24		27	220
25.0	30	150	27	330
26.0	27	680	30	200
27.0	27		30	270
28.0	51	62	30	430
29.0	33	240	30	910
30.0	30		33	330
31.0	33	510	39	150
32.0	47	100	33	1100
33.0	33		36	390
34.0	68	68	162	75
35.0	36	1200	39	330
36.0	36		39	470
37.0	39	750	43	270
38.0	39	1500	43	330
39.0	39		68	91
40.0	47	270	43	560
41.0	82	82	43	910
42.0	43	1800	62	130
43.0	43		47	510
44.0	47	680	62	150
45.0	82	100	47	1100
46.0	47	2200	51	470
47.0	47		82	110
48.0	51	820	56	330
49.0	56	390	51	1300
50.0	75	150	56	470
51.0	51		56	560
52.0	68	220	56	750
53.0	68	240	56	1000
54.0	56	1500	62	430
55.0	110	110	56	3000
56.0	56		75	220
57.0	75	240	62	680
58.0	91	160	62	910
59.0	62	1200	62	1300
60.0	68	510	75	300
62.0	62		75	360
64.0	68	1100	75	430
66.0	91	240	68	2200
68.0	68		91	270
70.0	82	470	75	1000
72.0	75	1800	120	180
74.0	75	5600	82	750
76.0	82	1000	91	470
78.0	62	1600	110	270
80.0	120	240	82	3300
82.0	82		91	620
84.0	91	1100	110	360
86.0	100	620	91	1500
88.0	120	330	91	2700
90.0	120	360	91	8200
92.0	110	560	120	390
94.0	100	1600	120	430
96.0	100	2400	110	750
98.0	100	5100	110	910

Parallel Resistance Calculator

This parallel resistance calculator provides an easy method for calculating the resistance of two resistors placed in parallel.

Although a parallel resistor value calculation is not difficult to calculate for two resistors placed in parallel, using this parallel resistor calculator simplifies the calculation rather than using paper and calculators.

Simply enter values of the parellal resistors in ohms, Ω or kΩ, etc in the two input boxes, but note all values must be in the same units. The parallel resistor calculator will then provide the overall resistance of the two resistors in the same units as the input.

Enter the two values for the resistors, R1 and R2, and the total resistance will be calculated for the parallel resistors.

Parallel Resistance Calculator

Enter Resistor Values:
R1:		ohms, Ω
R2:		ohms, Ω
Results:
Rtotal:		ohms Ω

Resistivity table / chart

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	~0.6 - 0.9 x 10^-7
Cadmium	6 x 10^-8
Cobalt	5.6 x 10^-8
Copper	1.7 x 10^-8
Gold	2.4 x 10^-8
Carbon (Graphite)	1 x 10^-5
Germanium	4.6 x 10^-1
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

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
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

Electrical Conductivity Basics

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

Temperature Coefficient of Resistance

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
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

Electrical Conductivity Basics

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

Capacitors and Capacitance Basics

Capacitance is one of the most important effects used in electronics. Along with this the associated components - capacitors are widely used, the second most widely used component.

Capacitors find uses in virtually every form of electronics circuit from analogue circuits including amplifiers and power supplies through to oscillators, integrators and many more. Capacitors are also used in logic circuits, primarily for providing decoupling to prevent spikes and ripple on the supply lines which could cause spurious triggering of the circuits.

What is capacitance

Capacitance is the ability to store electric charge. In its simplest form a capacitor consists of two parallel plates or electrodes that are separated from each other by an insulating dielectric. It is found that when a battery or any other voltage source is connected to the two plates as shown a current flows for a short time as it charges up. It is found that one plate of the capacitor receives an excess of electrons, while the other has too few. In this way the capacitor plate or electrode with the excess of electrons becomes negatively charged, while the capacitor electrode becomes positively charged.

Charge stored between two plates of a capacitor

If the battery is removed the capacitor will retain its charge. However if a resistor is placed across the plates, a current will flow in the resistor until the capacitor becomes discharged.

Units of capacitance

It is necessary to quantify a capacitor in terms of its ability to store charge. The basic unit of capacitance is the Farad, named after Michael Faraday.

The definition of A Farad is: A capacitor has a capacitance of one Farad when a potential difference of one volt will charge it with one coulomb of electricity (i.e. one Amp for one second).

In view of the fact that a capacitor with a capacitance of one Farad is too large for most electronics applications, components with much smaller values of capacitance are normally used. Three prefixes (multipliers) are used, µ (micro), n (nano) and p (pico):

PREFIX	MULTIPLIER
µ	10-6 (millionth)	1000000µF = 1F
n	10-9 (thousand-millionth)	1000nF = 1µF
p	10-12 (million-millionth)	1000pF = 1nF

Capacitor charge discharge cycle

It is also possible to look at the voltage across the capacitor as well as looking at the charge. After all it is easier to measure the voltage on it using a simple meter. When the capacitor is discharged there is no voltage across it. Similarly, one it is fully charged no current is flowing from the voltage source and therefore it has the same voltage across it as the source.

In reality there will always be some resistance in the circuit, and therefore the capacitor will be connected to the voltage source through a resistor. This means that it will take a finite time for the capacitor to charge up, and the rise in voltage does not take place instantly. It is found that the rate at which the voltage rises is much faster at first than after it has been charging for some while. Eventually it reaches a point when it is virtually fully charged and almost no current flows. In theory the capacitor never becomes fully charged as the curve is asymptotic. However in reality it reaches a point where it can be considered to be fully charged or discharged and no current flows.

Similarly the capacitor will always discharge through a resistance. As the charge on the capacitor falls, so the voltage across the plates is reduced. This means that the current will be reduced, and in turn the rate at which the charge is reduced falls. This means that the voltage across the capacitor falls in an exponential fashion, gradually approaching zero.

The rate at which the voltage rises or decays is dependent upon the resistance in the circuit. The greater the resistance the smaller the amount of charge which is transferred and the longer it takes for the capacitor to charge or discharge.

Voltage on a capacitor charging and discharging

So far the case when a battery has been connected to charge the capacitor and disconnected and a resistor applied to charge it up have been considered. If an alternating waveform, which by its nature is continually changing is applied to the capacitor, then it will be in a continual state of charging and discharging. For this to happen a current must be flowing in the circuit. In this way a capacitor will allow an alternating current to flow, but it will block a direct current. As such capacitors are used for coupling an AC signal between two circuits which are at different steady state potentials.

Phase

In an electric circuit it is found that the voltage and current are not exactly in phase. Because current flows through the capacitor when there is a change in voltage the current leads the voltage by 90 degrees. The maximum rate of change in voltage takes place when the voltage is midway between the two peaks. This is when the maximum current flows. The minimum rate of change of voltage occurs at either peak and hence the current is at a minimum.

Capacitor Equations

There are many calculations and equations associated with capacitors. The capacitor reactance equations and calculations are common, but there are many more capacitor calculations that may need to be performed.

Capacitor equations and capacitor calculations include many aspects of capacitor operation including the capacitor charge, capacitor voltage capacitor reactance calculations and many more.

Basic capacitance formulae

The very basic capacitor equations link the capacitance with the charge held on the capacitor, and the voltage across the plates.

where
    C is the capacitance in Farads
    Q is the charge held on the plates in coulombs
    V is the potential difference across the plates in volts

This equation can then be developed to calculate the work required for charging a capacitor, and hence the energy stored in it.

Capacitor reactance

In a direct current circuit where there may be a battery and a resistor, it is the resistor that resists the flow of current in the circuit. This is basic Ohms Law. The same is true for an alternating current circuit with a capacitor. A capacitor with a small plate area will only be able to store a small amount of charge, and this will impede the flow of current. A larger capacitor will allow a greater flow of current. A capacitor is said to have a certain reactance. This name is chosen to be different to that of a resistor, but it is measured in Ohms just the same. The reactance of a capacitor is dependent upon the value of the capacitor and also the frequency of operation. The higher the frequency the smaller the reactance.

The actual reactance can be calculated from the formula:

where
    Xc is the capacitive reactance in ohms
    f is the frequency in Hertz
    C is the capacitance in Farads

Current calculations

The reactance of the capacitor that is calculated from the formula above is measured in Ohms. The current flowing in the circuit can then be calculated in the normal way using Ohms Law:

Adding resistance and reactance

Although resistance and reactance are very similar, and the values of both are measured in Ohms, they are not exactly the same. As a result it is not possible to add them together directly. Instead they have to be summed "vectorially". In other words it is necessary to square each value, and then add these together and take the square root of this figure. Put in a more mathematical format:

where
    Xtot is the total impedance in ohms
    Xc is the capacitive reactance in ohms
    R is the DC resistance in ohms

Capacitor dielectric constant & permittivity

Permittivity and dielectric constant are two terms that are at the very heart of capacitor technology. The dielectric is the material that provides the insulation between the capacitor plates, and many of the characteristics of the capacitor will be dependent upon the properties of the dielectric used.

Capacitor permittivity and dielectric constant

The terms permittivity and dielectric constant are essentially the same for most purposes, although care must be taken when interesting some of the terms are relative permittivity and other terms have some specific meanings.

It is that property of a dielectric material that determines how much electrostatic energy can be stored per unit of volume when unit voltage is applied, and as a result it is of great importance for capacitors and capacitance calculations and the like.

In general permittivity uses the Greek letter epsilon as its symbol: ε.

Definitions of some specific terms related to dielectric constant and permittivity are given below:

• Absolute permittivity:   is the measure of permittivity in a vacuum and it is how much resistance is encountered when forming an electric field in a vacuum. The absolute permittivity is normally symbolised by ε0. The permittivity of free space - a vacuum - is equal to approximately 8.85 x 10-12 Farads / metre (F/m)
• Relative permittivity:   is permittivity of a given material relative to that of the permittivity of a vacuum. It is normally symbolised by: εr.
• Static permittivity:   of a material is its permittivity when exposed to a static electric field. Often a low frequency limit is placed on the material for this measurement. A static permittivity is often required because the response of a material is a complex relationship related to the frequency of the applied voltage.
• Dielectric constant:   This is the relative permittivity for a substance or material.

It can be seen from the definitions or permittivity that constants are related according toth e following equation:

εr   =   εs   /   ε0

Where:
εr = relative permittivity
εs = permittivity of the substance in Farads per metre
ε0 = permittivity of a vacuum in Farads per metre

Choice of capacitor dielectric

Capacitors use a variety of different substances as their dielectric material. The material is chosen for the properties it provides. One of the major reasons for the choice of a particular dielectric material is its dielectric constant. Those with a high dielectric constant enable high values of capacitance to be achieved - each one having a different permittivity or dielectric constant. This changes the amount of capacitance that the capacitor will have for a given area and spacing.

The dielectric will also need to be chosen to meet requirements such as insulation strength - it must be able to withstand the voltages placed across it with the thickness levels used. It must also be sufficiently stable with variations in temperature, humidity, and voltage, etc.

Relative permittivity of common substances

The table below gives the relative permittivity of a number of common substances.

SUBSTANCE	RELATIVE PERMITTIVITY
Ebonite	2.7 - 2.9
Glass	5 - 10
Marble	8.3
Mica	5.6 - 8.0
Paraffin wax	2 - 2.4
Porcelain	4.5 - 6.7
Rubber	2.0 - 2.3
Calcium titanate	150
Strontium titanate	200
Air 0C	1.000594
Air 20C	1.000528
Carbon monoxide 25C	1.000634
Carbon dioxide 25C	1.000904
Hydrogen 0C	1.000265
Helium 25C	1.000067
Nitrogen 25C	1.000538
Sulphur dioxide 22C	1.00818

The values given above are what may be termed the "static" values of permittivity. They are true for steady state or low frequencies. It is found that the permittivity of a material usually decreases with increasing frequency. It also falls with increasing temperature. These factors are normally taken into account when designing a capacitor for electronics applications. Some materials have a more stable level of permittivity and hence they are used in the higher tolerance capacitors. However this often has to be balanced against other factors. Some materials have very high levels of permittivity, and hence they enable capacitors to be made much smaller. This factor may be particularly useful when the size of the capacitor is particularly important.

Capacitor ESR, Dissipation Factor, Loss Tangent and Q

ESR - Equivalent Series Resistance, DF - Dissipation Factor, and Q or Quality factor are three important factors in the specification of any capacitor. They have a marked impact on the performance of the capacitor and can govern the types of application for which the capacitor may be used. As the three parameters are interlinked, ESR, DF and Q will all be addressed on this page.

ESR, DF and Q are all aspects of the performance of a capacitor that will affect its performance in areas such as RF operation. However ESR, and DF are also particularly important for capacitors operating in power supplies where a high ESR and dissipation factor, DF will result in large amount of power being dissipated in the capacitor.

Capacitor ESR, Equivalent Series Resistance

The equivalent series resistance or ESR of a capacitor is particularly important in many applications. One particular area where it is of paramount importance is within power supply design for both switching and linear power supplies. In view of the high levels of current that need to be passed in these applications, the equivalent series resistance, ESR plays a major part in the performance of the circuit as a whole.

The ESR of the capacitor is responsible for the energy dissipated as heat and it is directly proportional to the DF. When analysing a circuit fully, a capacitor should be depicted as its equivalent circuit including the ideal capacitor, but also with its series ESR.

The Equivalent Series Resistance, ESR associated with a capacitor

Capacitors with high values of ESR will naturally need to dissipate power as heat. For some circuits with only low values of current, this may not be a problem, however in many circuits such as power supply smoothing circuits where current levels are high, the power levels dissipated by the ESR may result in a significant temperature rise. This needs to be within the operational bounds for the capacitor otherwise damage may result, and this needs to be incorporated within the design of the circuit.

It is found that when the temperature of a capacitor rises, then generally the ESR increases, although in a non-linear fashion. Increasing frequency also has a similar effect.

Dissipation factor and loss tangent

Although the ESR figure of a capacitor is mentioned more often, dissipation factor and loss tangent are also widely used and closely associated with the capacitor ESR.

Although dissipation factor and loss tangent are effectively the same, they take slightly different views which are useful when designing different types of circuit. Normally the dissipation factor is used at lower frequencies, whereas the loss tangent is more applicable for high frequency applications.

The dissipation factor can be defined as: the value of the tendency of dielectric materials to absorb some of the energy when an AC signal is applied.

The loss tangent is defined as: the tangent of the difference of the phase angle between capacitor voltage and capacitor current with respect to the theoretical 90 degree value anticipated, this difference being caused by the dielectric losses within the capacitor. The value δ (Greek letter delta) is also known as the loss angle.

Capacitor loss tangent

Thus:

Where:
    δ = loss angle (Greek letter delta)
    DF = dissipation factor
    Q = quality factor
    ESR = equivalent series resistance
    Xc = reactance of the capacitor in ohms.

Capacitor Q

It is convenient to define the Q or Quality Factor of a capacitor. It is a fundamental expression of the energy losses in a resonant system. Essentially for a capacitor it is the ratio of the energy stored to that dissipated per cycle.

It can further be deduced that the Q can be expressed as the ratio of the capacitive reactance to the ESR at the frequency of interest:

As Q can be measured quite easily, and it provides repeatable measurements, it is an ideal method for quantifying the loss in low loss components.

Capacitor conversion chart / table

Capacitor conversion charts can be useful to convert capacitor values between picofarads, nanofarads and microfarads. Capacitors are available in an enormous range of values. The smallest capacitors used in electronics circuit designs may be only a picofarad or so, whereas the highest value capacitors that are commonly used may be as large as a few hundred or thousand microfarads.

This is a range of over 109. To prevent confusion with large numbers of zeros attached to the values of the different capacitors the common prefixes pico (10-12), nano (10-9) and micro (10-6 are widely used. When converting between these it is sometimes useful to have a capacitor conversion chart or capacitor conversion table for the different capacitor values.

Capacitor conversion table for picoFarads, nanoFarads, and microFarads

MICROFARADS (ΜF)	NANOFARADS (NF)	PICOFARADS (PF)
0.000001	0.001	1
0.00001	0.01	10
0.0001	0.1	100
0.001	1	1000
0.01	10	10000
0.1	100	100000
1	1000	1000000
10	10000	10000000
100	100000	100000000

This capacitor conversion chart or capacitor conversion table enables quick and easy reference of the different values given for capacitors and conversion between picofarads, nanofarads and microfarads.

Inductors & Inductance Basics

In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field.

Inductors generate an opposing voltage proportional to the rate of change in current in a circuit.

This property also is called self-inductance to discriminate it from mutual inductance, describing the voltage induced in one electrical circuit by the rate of change of the electric current in another circuit.

Inductance is one of the basic circuit parameters used in circuit design and development.

Inductors appear in a variety of formats, as chokes, transformers, inductors and many other items.

Inductance basics

Inductance is caused by the magnetic field generated by electric currents flowing within an electrical circuit. Typically coils of wire are used as a coil increases the coupling of the magnetic field and increases the effect.

There are two ways in which inductance is used:

• Self-inductance:   Self-inductance is the property of a circuit, often a coil, whereby a change in current causes a change in voltage in that circuit due to the magnetic effect of caused by the current flow.. It can be seen that self-inductance applies to a single circuit - in other words it is an inductance, typically within a single coil. This effect is used in single coils or chokes.
• Mutual-inductance:   Mutual inductance is an inductive effect where a change in current in one circuit causes a change in voltage across a second circuit as a result of a magnetic field that links both circuits. This effect is used in transformers.

Inductance circuit symbols

The circuit symbol for an inductor indicates the coil nature of the inductor. There are several formats indicating whether the inductor or transformer has a air core or a magnetic core.

Selection of inductor circuit symbols

While the basic inductor is widely used in many circuits, the transformer is also used in very many applications.

Selection of transformer circuit symbols

Inductance units

When indicating an inductor on a circuit diagram or within an equation, generally the symbol "L" is used. On circuit diagrams, inductors are generally numbered, L1, L2, etc.

The SI unit of inductance is the henry, H. The inductance of a circuit is one henry if the rate of change of current in a circuit is one ampere per second and this results in an electromotive force of one volt.

One henry is equal to 1 Wb/A.

Self Inductance

Inductance is defined as the magnetic induction of a voltage in a current carrying wire when the current in a wire changes. This can occur in the same wire and additionally in another wire.

In the case of self-inductance, the magnetic field created by a changing current in the circuit induces a voltage in the same wire or circuit - in other words any voltage is self-induced.

Self-inductance definition

Self-inductance can be defined as:

• the phenomenon in which a change in electric current in a circuit produces an induced electro-motive-force in the same circuit.

In terms of the units the following self-induction definition may be applied:

• The self-inductance of a coil is said to be one henry if a current change of one ampere per second through a circuit produces an electro-motive force of one volt in the circuit.

Self-inductance basics

When current passes along a wire, and especially when it passes through a coil or inductor, a magnetic field is induced. This extends outwards from the wire or inductor and could couple with other circuits. However it also couples with the circuit from which it is set up.

The magnetic field can be envisaged as concentric loops of magnetic flux that surround the wire, and larger ones that join up with others from other loops of the coil enabling self-coupling within the coil.

When the current in the coil changes, this causes a voltage to be induced the different loops of the coil - the result of self-inductance.

In terms of quantifying the effect of the inductance, the basic formula below quantifies the effect.

Where:
    VL = induced voltage in volts
    N = number of turns in the coil
    dφ/dt = rate of change of magnetic flux in webers / second

The induced voltage in an inductor may also be expressed in terms of the inductance (in henries) and the rate of change of current.

Lenz's law and self-induction

It can be seen from the formula that the voltage induced by a change in current is in the opposite sense to the change in current. Any current induced in a conductor will oppose the change in current that caused the change in flux.

This is effectively what Lenz's law states because an induced current has a direction such that its magnetic field opposes the change in magnetic field that induced the current.

Lenz's law states that an induced electromotive force, EMF gives rise to a current whose magnetic field opposes the original change in magnetic flux.

Inductive Reactance Formulae & Calculations

Any inductor will resist the flow of an alternating current due to its inductance.

Even an inductor with zero resistance will resist the flow of current in this way.

The degree to which the inductor impedes the flow of current is due to its inductive reactance.

Inductive reactance basics

The effect by which the current flow of an alternating or changing current in an inductor is reduced is called its inductive reactance. Any changing current in an inductor will be impeded as a result of the inductance associated with it.

The reason for this inductive reactance can be simply seen by examining the self-inductance and its effect within the circuit.

When a changing current is applied to an inductor, the self-inductance gives rise to an induced voltage. This voltage is proportional to the inductance and as a result of Lenz's law the induced voltage is in the opposite sense to the applied voltage. In this way the induced voltage will work against the voltage causing the current to flow and in this way it will impede the current flow.

Inductive reactance equations

When a changing signal such as a sine wave is applied to a perfect inductor, i.e. one with no resistance, the reactance impedes the flow of current, and follows Ohms law.

Where:
    XL = inductive reactance on ohms, Ω
    V = voltage in volts
    I = current in amps

The inductive reactance of an inductor is dependent upon its inductance as well as the frequency that is applied.

Where:
    XL = inductive reactance on ohms, Ω
    π = Greek letter Pi, 3.142
    f = frequency in Hz
    L = inductance in henries

R-L circuits and inductive reactance

In reality an inductor will have some resistance, and also inductors may be combined with resistors to make a combined network. As a result of the fact that the current and voltage within an inductor are 90° out of phase with each other (current lags the voltage), inductive reactance and resistance cannot be directly added.

As a result of the fact that the current and voltage in the inductor are out of phase, this means that the resistance and inductive reactance cannot be directly added together.

It can be seen from the diagram that the two quantities need to be added together vectorially. This means that the inductive reactance and resistance each need to be squared, added and then the resultant square root taken:

The resultant combination of resistance and inductive reactance is referred to as impedance and this is again measured in ohms.

Electronic Component Circuit Symbols

Circuit symbols are used for electronic circuit diagrams or circuit schematics. The various schematic symbols are used to represent different electronic components and devices in circuit diagrams from wires to batteries and passive components to semiconductors, logic circuits and highly complicated integrated circuits.

By using a common set of circuit symbols in schematics, it is possible for electronic engineers around the globe to communicate circuit information concisely and without ambiguity.

Although there are a number of different standards in use for the different circuit symbols around the globe, the differences are normally small, and because most systems are well known, there is normally little room for ambiguity.

Circuit symbol systems

There is a number of different systems used for schematic symbols around the globe. The main ones are listed below:

• IEC 60617:   This standard is issued by the International Electrotechnical Commission, and this standard for electronic component symbols is based on the older British Standard, BS 3939 which in turn was developed from the much older British Standard 530. Often reference is made to BS electric component standard, and the IEC standard is now the one that is used. The database includes around 1750 circuit symbols overall.
• ANSI standard Y32:   This standard for electronic component symbols is the American one and is also known as IEEE Std 315. This IEEE standard for circuit symbols has various release dates.
• Australian Standard AS 1102:   This is an Australian standard for electronic component symbols.

Of these the IEC and ANSI/IEEE standards for electronic symbols, i.e. schemtic symbols are those that are most widely used. Both are quite similar to each other although there are a number of differences. However as many circuit diagrams are used globally, both systems will be well known to most electronics engineers.

Circuit notation and reference designators

When developing a circuit diagram or schematic, it is necessary to identify the individual components. This is particularly important when using a parts list as the components on the circuit diagram can be cross related to the parts list or Bill of Materials. It is also essential to identify components as they are often marked on the printed circuit board and in this way the circuit and the physical component can be identified for activities such as repair, etc..

In order to identify components, what is termed a circuit reference designator is used. This circuit reference designator normally consists of one or two letters followed by a number. The letters indicate the type of component, and the number, defines which particular component of that type it is. An example may be R13, or C45, etc..

In order to standardise the way in which components are identified within schematics, the IEEE introduced a standard IEEE 200-1975 as the "Standard Reference Designations for Electrical and Electronics Parts and Equipments." This was later withdrawn and later the ASME (American Society of Mechanical Engineers), initiated the new standard ASME Y14.44-2008.

Some of the more commonly used circuit reference designators are given below:

MORE COMMONLY USED CIRCUIT SCHEMATIC REFERENCE DESIGNATORS
REFERENCE DESIGNATOR	COMPONENT TYPE
ATT	Attenuator
BR	Bridge rectifier
BT	battery
C	Capacitor
D	Diode
F	Fuse
IC	Integrated circuit - an alternative widely used non-standard abbreviation
J	Connector jack (normally but not always refers to female contact)
L	Inductor
LS	Loudspeaker
P	Plug
PS	Power supply
Q	Transistor
R	Resistor
S	Switch
SW	Switch - an alternative widely used non-standard abbreviation
T	Transformer
TP	Test point
TR	Transistor - an alternative widely used non-standard abbreviation
U	Integrated circuit
VR	Variable resistor
X	Transducer
XTAL	Crystal - an alternative widely used non-standard abbreviation
Z	Zener diode
ZD	Zener diode - an alternative widely used non-standard abbreviation

Circuit diagram symbols

There are very many circuit diagram symbols for different components. Accordingly the different symbols have been grouped into different sections which have been provided on the pages as set out below:

• Passive component circuit symbols:   This section includes many of the common passive component schematic symbols including resistors, capacitors and inductors, including transformers. Read more . . .
• Semiconductor diode symbols:   This section includes circuit symbols for the variety of forms of semiconductor diodes that are available. Read more . . .
• Transistor & FET circuit symbols:   This page includes schematic symbols for transistor and FET components as well as thyristor, DIAC and TRIAC. Read more . . .
• Wires, switches, connectors:   The circuit symbols for wires connectors, and switches. Read more . . .
• Logic circuit symbols:   Logic circuit blocks are widely used in digital electronics. These logic elements may be contained within an IC, but need to be depicted as separate elements to show the functionality of the circuit. Read more . . .
• Analogue functional building blocks:   Often circuit symbols are required for building blocks that are more than simple components. These blocks may be bought in as specific items, or they may be sued to give a block diagram of a system. Read more ...

  Passive Component Circuit Symbols

There are many different circuit symbols for passive components.

This page provides circuit symbols for passive components including resistors, capacitors, inductors and transformers.

As these circuit diagram symbols for some passive components, particularly resistors vary according to the geographical location where they were generated, those for US and European commonly used versions are given where possible.

The circuit symbols are based on those that appear to be most widely used.

Resistor circuit diagram symbols

There are many different circuit symbols for passive components.

This page provides circuit symbols for passive components including resistors, capacitors, inductors and transformers.

As these circuit diagram symbols for some passive components, particularly resistors vary according to the geographical location where they were generated, those for US and European commonly used versions are given where possible.

The circuit symbols are based on those that appear to be most widely used.

	Resistor (Europe)
	Potentiometer (Europe)
	Variable resistor (Europe)
	Resistor (US)
	Potentiometer (US)
	Variable resistor (US)
	Light dependent resistor, LDR
	Thermistor (temperature dependent resistor)

Capacitor circuit diagram symbols

The selection of capacitor schematic symbols:

	Capacitor (Fixed non-polar)
	Variable capacitor (Operator adjustable)
	Variable capacitor (Preset)
	Capacitor (Fixed polar Europe)
	Capacitor (Fixed polar US)

Inductor circuit diagram symbols

The selection of inductor schematic symbols:

	Inductor
	Inductor with ferrite core
	Inductor with laminated / iron core
	Inductor with tap on the coil (Note: the position of the tap can be moved to indicate approximately where on the coil the tap is positioned).

Transformer circuit diagram symbols

The selection of transformer schematic symbols:

	Transformer with air core (Note: turns on either side of the transformer may be alterd to indicate step up or step down).
	Transformer with ferrite core
	Transformer with laminated or iron core
	Transformer with centre tap
	Transformer with adjustable core (Typically used for tuning, e.g. as in IF transformers, etc).

Semiconductor Diode Component Circuit Symbols

Semiconductor diode circuit diagram symbols

The selection of semiconductor circuit diagram symbols:

	Basic diode
	Basic diode  (showing electrodes)
	Zener / voltage reference diode
	Varactor / varicap diode
	Schottky diode
	Tunnel diode
	Photodiode
	Light emitting diode, LED

Transistor & FET Component Circuit Symbols

There are many different circuit symbols for transistors, FETs, MOSFETs and SCRs, thyristors, TRIACs and DIACs, etc.

This page provides circuit symbols for transistors, FETs, MOSFETs and SCRs, thyristors, TRIACs and DIACs.

The circuit symbols are based on those that appear to be most widely used.

Basic bipolar transistor circuit symbols

The selection of transistor circuit diagram symbols:

	Bipolar transistor (NPN - this type is the more widely used variety)
	Bipolar transistor (PNP)
	Bipolar transistor (no circle around symbol as often seen)
	Bipolar transistor (with base, emitter and collector electrodes marked - this is generally only seen on basic circuits, i.e. for hobbyists etc)
	Bipolar phototransistor (with base not connected)
	Bipolar phototransistor

Darlington transistor circuit symbols

Darlington transistor circuit diagram symbols:

	Darlington transistor
	Photodarlington transistor
	Photodarlington transistor with base connection

FET circuit symbols

Darlington transistor schematic diagram symbols:

Dual gate MOSFET

SCR, TRIAC, & DIAC circuit symbols

SCR, TRIAC, & DIAC circuit diagram symbols:

	DIAC
	Thyristor (Silicon Controleld Rectifier, SCR)
	TRIAC
	TRIAC (without electrode names, etc)

Wires, Feeders, Switches, Connectors Circuit Symbols

There are many different circuit symbols for wires, feeders,connectors and switches, etc.

This page provides electronic component circuit symbols for wires, feeders, connectors and switches.

These electronic component circuit symbols are based on those that appear to be most widely used.

Wire circuit symbols

The selection of schematic symbols for wire:

	Crossed wires
	Crossed wires (alternative which is less standard these days).
	Joined wires
	Joined wires
	Coax feeder
	Shielding / screening
	Test point or terminal

Switch circuit symbols

The selection of circuit diagram symbols for switches:

	Switch (SPST)
	Switch (SPST - normally closed)
	Switch (SPST - push button / momentary action)
	Switch (DPST)
	Switch (SPDT)
	Switch (Single pole four way)

Note abbreviations for switches include:

    SPST: Single pole single throw
    DPST: Double pole, single throw
    SPDT: Single pole, double throw
    DPDT: Double pole, double throw

The number of "throws" equates to the number of active positions on the switch, and the number of poles equates to the number of circuits that are switched.

Digital / Logic Circuit Symbols

There is a vareity of different symbols for logic elements within a circuit. NAND, NOR and AND and OR are the most common circuit symbols, but oters are also used.

This page provides circuit symbols for logic elements.

The circuit symbols are based on those that appear to be most widely used.

Logic circuit symbols

The selection of logic circuit diagram symbols:

	Logic buffer
	Logic inverter
	Logic AND gate
	Logic NAND gate
	Logic OR gate
	Logic NOR gate
	Logic exclusive OR gate
	Logic exclusive NOR gate

Functional Blocks Circuit Symbols

Sometimes circuit symbols are needed for functional blocks within circuits.

These functional blocks may include elements such as filters, attenuators, mixers and other similar items.

The functional block circuit symbols are typically used when pre-manufactured items are bought and included within a circuit. Typically the circuit for the block would not be available, and as the item may be a sealed element, its circuit would not be required.

Analogue functional block circuit symbols

These analogue functional block circuit symbols have been defined in a number of specifications or standards including BS 3939. The circuit symbols used here are those that are typically or most commonly used:

	Fixed attenuator pad
	Variable attenuator pad
	Amplifying element / amplifier
	Mixer (multiplier / frequency changer)
	Mixer (alternative mixer circuit symbol)
	Filter (general filter circuit symbol)
	High pass filter
	Low pass filter
	Band pass filter
	Band reject filter

Quality Factor / Q Factor Tutorial

The quality factor or 'Q' of an inductor or tuned circuit is often used to give an indication of its performance in an RF or other circuit.

Values for quality factor are often seen quoted and can be used in defining the performance of an inductor or tuned circuit.

Accordingly this parameter is an important factor in the definition of various RF components and circuits.

Q, quality factor basics

The concept of quality factor is one that is applicable in many areas of physics and engineering and it is denoted by the letter Q and may be referred to as the Q factor.

The Q factor is a dimensionless parameter that indicates the energy losses within a resonant element which could be anything from a mechanical pendulum, an element in a mechanical structure, or within electronic circuit such as a resonant circuit. In particular Q is often used in association with an inductor.

While the Q factor of an element relates the losses, this links directly in to the bandwidth of the resonator with respect to its centre frequency. As such the Q or quality factor is particularly important within RF tuned circuits, filters, etc..

The Q indicates energy loss relative to the amount of energy stored within the system. Thus the higher the Q the lower the rate of energy loss and hence oscillations will reduce more slowly, i.e. they will have a low level of damping and they will ring for longer.

For electronic circuits, energy losses within the circuit are caused by resistance. Although this can occur anywhere within the circuit, the main cause of resistance occurs within the inductor. Accordingly inductor Q is a major factor within resonant circuits.

Effects of Q

When dealing with RF tuned circuits, there are many reasons why Q factor is important. Usually a high level of Q is beneficial, but in some applications a defined level of Q may be what is required.

Some of the considerations associated with Q in RF tuned circuits are summarised below:

• Bandwidth:   With increasing Q or quality factor, so the bandwidth of the tuned circuit filter is reduced. As losses decrease so the tuned circuit becomes sharper as energy is stored better in the circuit. 
  
  
  
  
  
  It can be seen that as the Q increases, so the 3 dB bandwidth decreases and the overall response of the tuned circuit increases.
• Ringing:   As the Q of a resonant circuit increases so the losses decrease. This means that any oscillation set up within the circuit will take longer to die away. In other words the circuit will tend to "ring" more. This is actually ideal for use within an oscillator circuit because it is easier to set up and maintain an oscillation as less energy is lost in the tuned circuit.
• Oscillator phase noise:   Any oscillator generates what is known as phase noise. This comprises random shifts in the phase of the signal. This manifests itself as noise that spreads out from the main carrier. As might be expected, this noise is not wanted and therefore needs to be minimised. The oscillator design can be tailored to reduce this in a number of ways, the chief one being by increasing the Q, quality factor of the oscillator tuned circuit.
• General spurious signals:   Tuned circuits and filters are often used to remove spurious signals. The sharper the filter and the higher the level of Q, the better the circuit will be able to remove the spurious signals.
• Wide bandwidth:   In many RF applications there is a requirement for wide bandwidth operation. Some forms of modulation require a wide bandwidth, and other applications require fixed filters to provide wide band coverage. While high rejection of unwanted signals may be required, there is a competing requirement for wide bandwidths. Accordingly in many applications the level of Q required needs to be determined to provide the overall performance that is needed meeting requirements for wide bandwidth and adequate rejection of unwanted signals.

Quality factor definition

The definition of quality factor is often needed to give a more exact understanding of what this quantity actually is.

For electronic circuits, Q is defined as the ratio of the energy stored in the resonator to the energy supplied by a to it, per cycle, to keep signal amplitude constant, at a frequency where the stored energy is constant with time.

It can also be defined for an inductor as the ratio of its inductive reactance to its resistance at a particular frequency, and it is a measure of its efficiency.

Q factor equations

The basic Q or quality factor equation is based upon the energy losses within the inductor, circuit or other form of component.

From the definition of quality factor given above, the Q factor can be mathematically expressed as:

When looking at the bandwidth of an RF resonant circuit this translates to:

Q of a tuned circuit with respect to its bandwidth

Within any RF or other circuit, each individual component can contribute to the Q or quality factor of the circuit network as a whole. The Q of the components such as inductors and capacitors are often quoted as having a certain Q or quality factor.

Quality factor and damping

One aspect of Q that is of importance in many circuits is the damping. The Q factor determines the qualitative behaviour of simple damped oscillators and affects other circuits such as the response within filters, etc.

There are three main regimes which can be considered when referring to the damping and Q factor.

• Overdamped (Q < 1/2):   An over-damped system is one where the Q factor is less than 1/2. In this type of system, the losses are high and the system has no overshoot, but instead the system will exponential decay, approaching the steady state value asymptotically after a step impulse is applied. As the quality factor is reduced, so the systems responds more slowly to a step impulse.
• Underdamped (Q > 1/2) :   An under-damped system is one where the Q factor is greater than a half. Those systems where the Q factor is only just over a half may oscillate once or twice when a step impulse is applied before the oscillation falls away. As the quality factor increases, so the damping falls and oscillations will be sustained for longer. In a theoretical system where the Q factor is infinite, the oscillation would be maintained indefinitely without the need for adding any further stimulus. In oscillators some signal is fed back to provide an additional stimulus, but a high Q factor normally produces a much cleaner result. Lower levels of phase noise are present on the signal.
• Critically damped (Q = 1/2) :   Like an over-damped system, the output does not oscillate, and does not overshoot its steady-state output. The system will approach the steady-state asymptote in the fastest time without any overshoot.

When choosing defining the Q factor for a system, it is common to opt for the highest level. In this way the optimum performance is normally achieved. However there are instances where lower levels of Q may be advantageous.

Inductor Q, Quality Factor

Even though inductors are often assumed to be pure inductors, they always have a finite amount of resistance, however low.

This DC resistance affects the inductor Q quality factor, and is one of the major factors affecting this area of performance of the component.

In view of this the inductor quality factor is widely specified for inductors to be used in RF applications.

Inductor Q factor basics

When using an inductor in a circuit where the Q or quality factor is important its resistance becomes an important factor. Any resistance will reduce the overall inductor Q factor.

An inductor can be considered in terms of its equivalent circuit. This can be simply expressed as a perfect inductor with a series resistor.

Where:
    L is a perfect inductor
    R is the resistance of the inductor

The resistance within an inductor is caused by a number of effects:

• Standard DC resistance:   The standard DC resistance will always be present (except in superconductors which are not normally encountered). This is one of the major components of resistance in any coil or inductor and one that can sometimes be reduced. Thicker wires, and sometimes silver or silver plated wires may be used.
• Skin effect:   The skin effect affects the inductor Q because it has the effect of raising the resistance. The skin effect results from the tendency of an alternating current flow through the outer areas of a conductor rather than through the middle. This has the effect of reducing the cross sectional area of the conductor through which the current can flow, thereby effectively increasing its resistance. It is found that the skin effect becomes more pronounced with increasing frequency.
  
  To reduce the effects of the skin effect different types of wire can be used:
  
  
  • Silver wire:   Silver or even silver plated wire can be used to reduce the effects of the skin effect. When compared to copper wire, silver wire has a lower resistance for a given surface area. To reduce the cost, silver plated wire can be used as the silver will be on the outside of the wire where most of the RF or alternating current is carried.
  • Litz wire:   Another form of wire that can be used is known as Litz wire. The name comes from the German word Litzendraht meaning braided, stranded or woven wire. It is a form of wire that consists of many thin strands of wire, each individually insulated and then woven together. In this way the surface area of the wire is considerably increased, thereby reducing the resistance to RF or alternating currents. Typically Litz wire is used for frequencies above about 500kHz, but below around 2 MHz.
• Radiated energy:   When an alternating current passes through an inductor, some of the energy will be radiated. Although this may be small, it still adds to the losses of the coil and in exactly the same way as occurs in an antenna this is represented by a radiation resistance. Accordingly this is a component of the inductor resistance and will reduce the inductor Q factor.
• Core losses:   Many inductors have ferrite or other forms of core these can introduce losses:
  
  
  • Eddy currents:   It is a commonly known fact that eddy currents can flow in the core of an inductor. These are currents that are induced within the core of the inductor. The eddy currents dissipate energy and mean that there are losses within the inductor which can be seen as an additional level of resistance that will reduce the inductor Q factor.
  • Hysteresis losses:   Magnetic hysteresis is another effect that causes losses and can reduce inductor Q factor values. The hysteresis of any magnetic material use as a core needs to be overcome with every cycle of the alternating current and hence the magnetic field. This expends energy and again manifests itself as another element of resistance. As ferrite materials are known for hysteresis losses,, the effect on the inductor quality factor can be minimised by the careful choice of ferrite or other core material, and also ensuring that the magnetic field induced is within the limits of the core material specified.

Minimising the resistance effects reduces the losses and increases the inductor Q factor.

Inductor Q factor equations

In order to calculate the Q, quality factor for an inductor, the equation or formula below can be used:

As Ω is equal to 2⋅π⋅f⋅L, this can be substituted in the equation to give:

From these equations it can be seen that it can be seen that the inductive reactance, X, varies with frequency. Accordingly the Q will also vary. In addition to this the resistive losses including those due to the skin effect, radiation losses, eddy current, and hysteresis, also vary with frequency and so will the inductor Q factor.

As a result the frequency of operation or measurement must be given for any inductor Q factor value.

Tuned Circuit Filter Quality Factor

Resonant circuits consist of inductors and capacitors. It is therefore often necessary to look at the quality factor of an LC, i.e. inductor capacitor system.

It is necessary to be able determine the LC filter Q factor to assess the performance of these circuits.

Fortunately there are some simple formulae or equations that can be used to determine the LC filter quality factor.

Q factor and LCR tuned circuits

One of the key features of an LC tuned circuit is that at resonance the inductive and capacitive reactances become equal. However dependent upon the type of tuned circuit, the effect is slightly different.

There are two basic types of tuned circuit:

• Parallel tuned circuit:   At resonance the impedance of a parallel tuned circuit peaks, decreasing either side of resonance. Below resonance the inductive reactance dominates and above resonance it becomes capacitive. As a result of its action any alternating or RF signal voltage placed across the circuit will peak at resonance.
• Series tuned circuit:   The series tuned circuit is very much the inverse of the parallel tuned circuit in that rather than showing a peak in impedance at resonance there is a minimum.

The equivalent circuit for a series tuned network is given below. In this, the resistance "R" is the equivalent series resistance for the inductor and capacitor:

A parallel tuned circuit is also affected by the resistance in the circuit.

.

In the case of the parallel tuned LC circuit, the Q factor is still an issue. Again there is resistance within both the inductor and capacitor. However as the inductor resistance dominates normally, it has been included in this leg for convenience.

LC Q factor equations

When determining the Q of an LC tuned circuit it is necessary to determine whether the circuit is series or parallel tuned. The LC Q factor for a series tuned circuit is:

The LC Q factor for a parallel tuned circuit is: