A thin metallic wire loop of arbitrary curvature is rotated with respect to an arbitrary axis of its plane. The device is excited by an electric dipole of infinite length and constant current. The resistance of the loop is computed rigorously as function of the position of the source. In this way, the induced voltage along the wire, under any kind of axial excitation, is given in the form of a superposition integral. The measured response is represented for various shapes of the coil, with respect to the time, the rotation angle and the position of the source. These diagrams lead to several technically applicable conclusions which are presented, discussed and justified.
Electromagnetic (EM) induction is the production of voltage across a conductor situated in a changing magnetic field or a conductor moving through a stationary magnetic field. The physics that govern the inductive experiments have been mathematically examined in a number of old and elementary studies. The relationship between the various induction laws is summarized in [
Many solving techniques have been utilized to convert the aforementioned theoretical analyses into practical applications in real-world configurations. For example in [
Apart from the old standard textbooks and the newer technical reports, there are numerous recent, state-ofthe-art references examining devices and presenting techniques which exploit electromagnetic induction. In [
In this work, we assume a closed metallic wire of arbitrary shape which is mechanically rotated with respect to an arbitrary axis of its plane, in the presence of an elemental source of infinite length and constant current. The parametric equation of the wire curve is extracted through coordinate transformations and the magnetic flux through the loop is evaluated straightforwardly. To this end, we apply the Faraday’s law and an expression for the coil’s resistance is derived. This study examines the voltage induction to the most general case of a single wire loop (shape and rotation axis can be altered at will) which has not been treated rigorously yet. However, the basic novelty of the present analysis is that, due to the linearity of the participating operators, the calculus can be generalized to cover any case of axial excitation. We compute a kind of “Green’s function” ([
The Cartesian coordinate system defined in
respect to the aforementioned axis. Initially, the curve lies entirely on the plane.
The purpose of this work is to obtain an expression for the electromotive force induced at the coil, being dependent on the time, the rotation axis and the shape of the loop. The generated voltage should certainly vary with the position of the primary source, and thus the derived formula could be used in studying structures with more complicatedly distributed excitation current.
Firstly, we should extract the parametric equation set of the rotated loop when as appeared in
and the azimuthal angle will play the role of the parametric variable even when the object does not belong exclusively to plane. As the closed wire is rotated with respect to axis, the corresponding coordinate will be fixed, independent from the angle and equal to. The rest two equations are derived by projecting the other edge of length, which is posed at angle, upon the axes and. Accordingly, one obtains the following expressions:
In order to find the parametric equation of the curve when, we use the primed coordinate system which is received through rotation of the unprimed one by angle with respect to axis (defined in
The parametric equations of the arbitrarily rotated wire loop expressed in the unprimed coordinate system are denoted by and are determined from the transformation relation below [
An extra time argument has been added to the dependencies of the parametric equations for better comprehension.
Once the boundary of the rotating wire is rigorously specified, the field quantities related to the induced voltage can be computed. The magnetic vector potential of a -polarized infinite dipole into vacuum at distance from its axis is given by [
where is the reference distance, is the vacuum magnetic permeability and is the unitary vector of the axis. The transverse distance of the representative loop point with azimuthal angle, at an arbitrary time, from the axis, is found apparently equal to:
also shown in
,(10)
where the subscript notation is used for the partial derivative of a function. It should be noted that the magnetic flux is independent from the reference distance. According to Faraday’s law, the induced electromotive force is given by the time derivative of magnetic flux; therefore, it is sensible to define the following auxiliary resistance function by excluding from (10):
In this way, the generated voltage on the coil due to the self-rotation, in the presence of a dipole posed along the axis, is given by:
In case the structure is excited by an axial surface current distributed according to the law (measured in), along the line, the induced electromotive force on the wire loop, is expressed in terms of the following line integral:
Such an expansion is permissible because the operator applied on to find, is linear. Similarly, if the excitation is a volume axial current (measured in), across the area, the voltage is obtained through the double integral below:
4. Indicative Results In this section, we examine the produced voltage by rotating loops of variable shape and the effect of their geometrical parameters on it. The described method is applied to two families (A, B) of wire boundaries possessing the following polar equations:
,(15)
The arguments have length dimension and the argument is a positive integer number. For brevity, we chose to excite the structure exclusively through point sources avoiding surface or volume distributions described through (13), (14). The following graphs will exhibit the dependencies of the produced voltage and the corresponding RMS value:
A single wire loop is not practically used for voltage generation as clusters of coils are utilized instead. We are more concerned for the qualitative description of the output quantity than measuring the exact magnitude (in Volts) of the produced voltage which is extremely small. Accordingly, the so-called “normalized voltage” is represented alternatively which is evaluated by replacing the magnetic permeability of the vacuum in (11), by unity. Each figure below is divided into two subfigures, the first of which is a diagram depicting the variation of the investigated quantities for various shapes of the loop, while the second is a polar plot of the corresponding wire boundaries.
In
In
In
these of
In
One could point out that the positive effect of the coil’s size on the induced measured quantity is greater when the excitation field is stronger (larger differences between the three curves in the vicinity of). It is also remarkable that the induced voltage is exponentially dependent on the distance between the thin closed wire and the source current, which is indicated not only by the decrease for large in
The case of a metallic loop of arbitrary shape rotated with respect to an arbitrary axis in the presence of an infinite dipole is investigated. The induced voltage is evaluated rigorously via transformation relations and the derived formula can be easily generalized to treat any kind of axial excitation. The time dependence of the measured voltage is of sinusoidal form only if the shape of the loop combined with the rotation axis is symmetric. Additionally, the intuitive guess that the size of the loop, affects positively the inductive action is verified by the numerical results. The distance between the source and the loop is also found to reduce the measured voltage along the coil. As far as the rotation axis is concerned, the most substantial response has been recorded when it is normal to the axis of the dipole current, a result that can have technical applicability.