The problem of finding the alternating-current resistance of single layer coils has been analysed by Butterworth, in a series of very complicated papers. It is shown here that by taking a different approach the analysis is relatively easy and produces simple equations which agree very well with published experimental measurements. The analysis presented here includes coils wound with flat conducting tape as well as conductors of circular cross-section.
The resistive loss of variable air capacitors is difficult to measure because they have a very high Q. The method described here uses a twin-wire transmission line made from copper pipe as the inductor to tune-out the reactance of the capacitor. Previous authors have shown that the capacitor loss comes partly from resistive loss and partly from dielectric loss in the insulating supports. They have attributed the resistive loss to that in the rotating contacts, but it is shown that there is also a major contribution from the metallic part of the structure, although the plates themselves have negligible resistance.
The inductance of a coil can be derived from the magnetic reluctance to its flux, and for a single layer coil this reluctance can be derived from simple equations for capacitance. These equations can then be used to solve the more difficult problem of the inductance of a coil when a ferrite rod is inserted.
When a ferrite rod is inserted into an air coil its inductance increases by a large factor, but the widely quoted equations for predicting the new inductance are shown to be flawed. A new theory is presented, based upon the magnetic reluctance, and this gives accurate predictions compared to experiment. Interestingly this shows that the increase in inductance when the ferrite is introduced is independent of the number of turns or their spacing or the inductance of the original air coil. Also if the ferrite permeability is high the increase is dependent only on the overall physical dimensions of the coil and ferrite.
The accepted theory for the radiation resistance of ferrite rod antenna is based on the theory of demagnetisation. This is shown to be flawed and a new theory is given which agrees well with experiment.
The losses in ferrite rod antennas are much higher than predicted by the accepted theory. This article shows that the increased loss is due to an increase in the copper losses in the winding rather than losses in the ferrite itself.
All coils show a self-resonant frequency, and as this frequency is approached the inductance and resistance increase while the Q decreases until a frequency is reached where the coil resonates in a similar way to a tuned circuit. Because of this similarity the effect has been attributed to self-capacitance in the coil, and many researchers have tried to reduce this capacitance in order to raise the Q. However nowhere in the coil can this capacitance be measured or deduced, and in fact the rising inductance and loss are explained if the coil is seen as a helical transmission line. This article discusses these issues, and gives accurate equations for the changes with frequency.
Litz cable can have a lower ac resistance than solid wire of the same overall diameter, but conventional wisdom says that it loses this advantage above a frequency of about 1 MHz. However it is shown here that Litz can have the lower resistance up to at least 14 MHz with readily available cable.
The high frequency resistance of two parallel wires presents serious theoretical difficulties, and in this article the theories are reviewed and experiments carried-out to determine their accuracy. It is then shown that the resistance of many wires in parallel can be calculated from the same equations, and that this sheet of wires is a good approximation to the difficult problem of the resistance of rectangular and strip conductors.
At high frequencies current in a conductor flows mainly on its surface, and this is known as the skin effect. Two possible mechanisms are given in the published literature, the first being that it is due to penetration of the conductor by an electromagnetic wave, and the second is that it is due to diffusion. This article resolves the issue by experiment, and shows that diffusion is the correct explanation. This has important implications in the screening of electronic circuits since many authorities in this area assume that the conductor is penetrated by an EM wave.
All coils have a self-resonant frequency (SRF), and as this frequency is approached the inductance and resistance increase while the Q decreases until a frequency is reached where the coil resonates in a similar way to a parallel tuned circuit. The coil thus appears to have self-capacitance and if this can be determined the changes in inductance with frequency can be readily calculated. This article derives equations for the self-capacitance of toroidal coils with ferrite cores.
Inductance coils are often wound around an insulating former which then provides mechanical support. The magnetic field is not affected by this former and so the low frequency inductance is not changed. However the inductance of all coils increases with frequency due to self-resonance, and the presence of the insulating former will increase this change of inductance because it lowers the self-resonant frequency (SRF). Losses in the former will reduce the Q of the coil, but interestingly it is found that the biggest reduction in Q is due to the reduction in the SRF. So a material with a low dielectric constant is preferable to one with a high dielectric constant, even if the loss factor of the low dielectric material is higher.
It is known that thin metal films can provide very good shielding of alternating magnetic fields, even at low frequencies. The mechanism is often thought to be due to eddy currents since these are known to produce a magnetic field which opposes the incident field, but no accurate theory is available. A theoretical analysis is given here and is shown to give excellent agreement with independent published measurements, including those of wire mesh and magnetic materials such as steel.
A simple equation is given here for the ac resistance of rectangular conductors. This is shown to give good agreement with independent measurements for all shapes of rectangular conductor from very thin strip to square section.
To detect the presence of a train on a railway track an AC current is passed down the rails. The effectiveness of this system depends on the resistance and inductance of the rails but the analysis is difficult because the rails are made from magnetic steel whose permeability and loss change with frequency, and with the amplitude of the current. A further difficulty at high frequencies is the complicated shape of the rail cross-section. This report models the rail as having a rectangular shape and then derives equations for the resistance and inductance and these agree well with published measurements.
When the magnetic field from one conductor cuts a second conductor currents are induced which give an increase in power loss. This loss must come from the conductor producing the field and so its resistance increases. Equations are given here and these agree well with measurements.
Eddy currents are induced in a conductor when there is an incident magnetic field. These currents produce an opposing magnetic field so that the resultant field is then smaller than the incident field. But how much smaller? What is the effect of the resistivity of the conductor and its magnetic permeability ? How does the resultant field vary with frequency ? These questions are answered here.
The magnetic permeability of a steel plate can be determined over a wide frequency range from a measurement of its skin depth. For this a current is established on one surface of the plate and the current which diffuses to the other surface is measured. The surface current can be induced by a magnetic field or by passing a current through two probes on the surface.
All coils show a self-resonant frequency (SRF), and as this frequency is approached the
inductance and resistance increase while the Q decreases until a frequency is reached
where the coil resonates in a similar way to a parallel tuned circuit. If the coil has an
open ferrite core, such as with a ferrite antenna, then the SRF is very much reduced, and
is dependent upon the permeability and permittivity of the ferrite.
Equations are given here for the resistance and inductance of traction rails at power frequencies and these equations agree well with published measurements. However it was found that for this good agreement an unrealistic B/H curve was needed of the steel, and this could be explained if the rails had a surface layer of lower resistivity and lower permeability, consistent with decarburisation of the steel.