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The property of inductance might be described as "when any piece of wire is wound into a coil form it forms an inductance which is the property of opposing any change in current". Alternatively it could be said "inductance is the property of a circuit by which energy is stored in the form of an electromagnetic field".
We said a piece of wire wound into a coil form has the ability to produce a counter emf (opposing current flow) and therefore has a value of inductance. The standard value of inductance is the Henry, a large value which like the Farad for capacitance is rarely encountered in electronics today. Typical values of units encountered are millihenries mH, one thousandth of a henry or the microhenry uH, one millionth of a henry.
A small straight piece of wire exhibits inductance (probably a fraction of a uH) although not of any major significance until we reach UHF frequencies.
The value of an inductance varies in proportion to the number of turns squared. If a coil was of one turn its value might be one unit. Having two turns the value would be four units while three turns would produce nine units although the length of the coil also enters into the equation.
The standard inductance formula for close approximation  imperial and metric is:
imperial measurements
L = r^{2} X N^{2} / ( 9r + 10len )
where:
L = inductance in uH
r = coil radius in inches
N = number of turns
len = length of the coil in inches
metric measurements
L = 0.394r^{2} X N^{2} / ( 9r + 10len )
where:
L = inductance in uH
r = coil radius in centimetres
N = number of turns
len = length of the coil in centimetres
[ADDED 22nd May, 2002] Someone asked about a formula which takes into account the spacing bewtween windings, the 10len above automatically takes that into account, if you're confused think about it!.
It has been found that the optimum dimensions for a high "Q" air core inductor is where the length of the coil is the same as the diameter of the coil. A simplified formula for inductance has been derived to establish the required number of turns for a given inductance value.
metric measurements
N = SQRT [( 29 * L ) / (0.394r)]
where:
L = inductance in uH
r = coil radius in centimetres
N = number of turns
Coils wound on a former (with or without a core) may have multilayers of windings which are called solenoid windings.
All coils also exhibit a degree of selfcapacitance caused by minute capacitances building up around and between adjacent windings.
Depending upon the application this may be of considerable concern. This selfcapacitance combined with the natural inductance will form a resonant circuit (selfresonant frequency) limiting the useful upper frequency of the coil. There are special winding techniques to to use on occassion to minimise this self capacitance.
If the coil is wound on an iron core the inductance is greatly increased and the magnetic lines of force increase proportionally. This is the basis of electromagnets used in solenoid valves and relays.
When the coil is wound on special iron laminations or cores and a second winding is placed on the core a "transformer" results. This is the basis of all power transformers although only alternating current (a.c.) can be transformed. The voltage relationship in transformers is proportional to the turns. For example a power transformer might have 2,500 turns on the primary side and the secondary side might have 126 turns. Such a relationship is 250 : 12.6 and if the primary were connected to 250V a.c. the secondary would produce a voltage of 12.6V a.c.
Interesting, if the core size and the wire diameter on the primary supported a primary current of 100 mA, the the primary power available would be 250V X 100 mA or 250 X 0.1 = 25 watts. Ignoring core and copper losses we could say that 25 watts is now available on the secondary side at 12.6V which is 25W / 12.6V = 1.98 amps. In practice we don't get that kind of efficiency however it would pay to remember that most power transformers are designed to function most efficient at or near full design load.
In many radio applications the coil is wound on a ferrite or powdered iron core. Typical examples are the ferrite rod receiving antenna used in cheap transistor radios or the i.f. transformers enclosed in metal cans in those radios  red, yellow, black, green cores. The core is manufactured to be optimum for the frequency range of interest and greatly enhances the inductance for a specific number of turns. If we wound a coil on a blank former we might get an inductance of say 10 uH, adding a specific core might increase the inductance to 47 uH. By using screw in / screw out cores (as in the metal cans) we can vary the inductance over a fair range of interest.
A home made Coil Winder by Lloyd Godsey KK7IZ  PDF File 473kB
BOOK  Inductor Handbook by Cletus J. Kaiser
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1. audio transformers
2. chokes
3. coil formers and cores
4. inductive reactance
5. mobius winding techniques
6. toroid cores
7. wide band rf transformers
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Updated 22nd May, 2002