Unipolar induction

Unipolar induction describes the separation of electrical charges with the help of the magnetic part of the Lorentz force and the associated creation of an electrical voltage . Although both DC and AC voltages can be generated using unipolar induction , its main application is in the generation of DC voltages. A typical arrangement consists of a circular, electrically conductive disk that rotates in a magnetic field parallel to its axis ( unipolar machine ).

Unipolar induction in a conductor loop

description

Moving conductor in a magnetic field; from the point of view of the laboratory system, there is no induction. The voltage that occurs is a potential difference.

The figure opposite shows a particularly simple imaginary arrangement in which unipolar induction occurs. The conductor rod moves with the speed in a temporally and spatially constant magnetic field with the flux density . The ends of the conductor bar are connected to metal rails, at the end of which the drawn voltage can be measured. ${\ displaystyle v}$ ${\ displaystyle {\ vec {B}}}$ ${\ displaystyle U}$

For the following description it is assumed that the observer is located in the laboratory system in which the metallic rails rest:

In the conductor rod, due to the magnetic component of the Lorentz force, a force acts on the electrons (charge of the electron ), which points "downwards" due to the negative charge of the electrons. The Lorentz force enables an electric current to flow when the circuit is closed . ${\ displaystyle {\ vec {F}} _ {L} = q {\ vec {v}} \ times {\ vec {B}}}$ ${\ displaystyle q = -e}$

In the illustrated open circuit, however, no electrical current can flow in the steady state. Thus, the Lorentz force cannot be the only force that acts on the electrons. The observer in the laboratory system therefore concludes that in addition to the Lorentz force, a Coulomb force must be present in the moving metallic conductor , which points “upwards” in the conductor rod and compensates for the Lorentz force. He explains the Coulomb force through a previous charge separation of the electrons. ${\ displaystyle {\ vec {F}} _ {C} = - q {\ vec {v}} \ times {\ vec {B}}}$

In an ohmically closed conductor circuit, the relationships are somewhat more complicated because the magnetic field changes over time due to the current flow and the movement of the conductor. In many practical arrangements, however, the inductance of the arrangement is very small, so that the flux density changes can be neglected.

Is there an electrical vortex field?

In the following, the obvious question should be clarified whether the voltage measured on the voltmeter is caused by vortex fields with closed electrical field lines. ${\ displaystyle U}$

The law of induction

{\ displaystyle \ oint \ limits _ {\ partial A} {\ vec {E}} \ mathrm {d} {\ vec {s}} = - \ int \ limits _ {A} {\ frac {\ partial {\ vec {B}}} {\ partial t}} \ mathrm {d} {\ vec {A}}}

describes how electrical vortex fields arise with the change in magnetic flux density over time . Since the magnetic flux density is constant over time for the arrangement described, this applies in the present case ${\ displaystyle {\ vec {B}}}$

{\ displaystyle {\ frac {\ partial {\ vec {B}}} {\ partial t}} = 0.}

As a result, no electrical vortex fields arise when the conductor rod moves uniformly.

Although the above argumentation is factually correct and mathematically easy to verify, it still seems at first glance to lead to an insurmountable contradiction, which can be described as follows:

In the arrangement described, an electrical voltage can be measured at the terminals between the rails . As a result, there is an electric field strength between the clamps (in the air) . ${\ displaystyle U}$ ${\ displaystyle {\ vec {E}} \ neq 0}$
However, if you connect the terminals in thought via a path that follows the metallic conductor and the conductor rod, there is no electrical field along this path. After all, metallic conductors are virtually free from electrical fields.
If you combine the routes described under 1 and 2 to form a closed loop route , the following applies to this: ${\ displaystyle s}$

{\ displaystyle \ oint \ limits _ {s} {\ vec {E}} \ mathrm {d} {\ vec {s}} \ neq 0}

As a result, there seem to be closed field lines after all.

The apparent contradiction can be resolved with the help of the special theory of relativity . The main mistaken assumption that leads to the apparent contradiction is that the metallic conductor rod (assumed to be ideally conductive) is free of electrical fields. In fact, however, the electric field strength is fundamentally dependent on the reference system in which it is measured. In reality, metallic conductors can only be assumed to be approximately field-free in reference systems from which the conductor is at rest. If one transforms the electric field strength in the moving conductor rod with the help of the Lorentz transformation into the rest system (laboratory system) noted without a line, one recognizes that the resting observer in the conductor rod measures an electrical field different from zero, which was made plausible in the introductory explanation about the Lorentz force . From the point of view of the laboratory system, the conductor bar therefore contains an electrical field that compensates for the voltage (also measured in the laboratory system) . ${\ displaystyle {\ vec {E}} '= 0}$ ${\ displaystyle {\ vec {E}} \ approx {\ vec {E}} '- {\ vec {v}} \ times {\ vec {B}} = - {\ vec {v}} \ times {\ vec {B}}}$ ${\ displaystyle U}$

Unipolar induction in the Faraday disk

Unipolar generator: An electrically conductive, rotating disk is located between the poles N and S of a strong permanent magnet . Between the center of rotation of the disk (axis) and the current collector on the circumference of the disk, a direct voltage can then be tapped as a result of the unipolar induction and displayed on the measuring device.

Calculation taking into account the Lorentz force

The linear arrangement with the moving conductor bar is not suitable for generating a direct voltage, since the conductor bar would have to move further and further away from the terminals over time. Instead, a cylinder-symmetrical arrangement similar to the Faraday disk shown on the right is recommended.

The terminal voltage on the Faraday disk is based - just as in the example with the moving conductor bar - on the Lorentz force on the charge carriers in the rotating body. It is assumed that the disk rotates around its axis at angular velocity in a homogeneous, axially parallel magnetic field . A voltage is measured between the axis and a sliding contact at a distance from the axis. ${\ displaystyle \ omega}$ ${\ displaystyle B}$ ${\ displaystyle R}$

The Lorentz Force

{\ displaystyle {\ vec {F}} _ {L} = q \, {\ vec {v}} \ times {\ vec {B}}}

on the conduction electrons, which rotate with the disk, is in equilibrium with the field force in the electric field generated by the charge separation

{\ displaystyle {\ vec {F}} _ {C} = q {\ vec {E}}}

${\ displaystyle {\ vec {B}}}$ : Vector of magnetic flux density
${\ displaystyle {\ vec {v}}}$ : Speed vector
${\ displaystyle e}$ : Elementary charge.

As perpendicular to is when the magnetic field passes through the wheel normal, the equilibrium of forces applies , ie . ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {B}}}$ ${\ displaystyle {\ vec {F}} _ {L} + {\ vec {F}} _ {C} = {\ vec {0}}}$ ${\ displaystyle {\ vec {E}} = - {\ vec {v}} \ times {\ vec {B}}}$

According to the amount we get:

{\ displaystyle q \, E = q \, v \, B}

{\ displaystyle q \, E = q \, \ omega \, r \, B}

${\ displaystyle r}$ : Distance of the electron from the axis of rotation
${\ displaystyle \ omega}$ : Angular velocity of the disk
${\ displaystyle E}$ : Field strength of the electric field corresponding to the Lorentz force.

Integrating E (r) results in the induction voltage between the central axis and the edge of the disk with radius R:

{\ displaystyle U = \ int \ limits _ {0} ^ {R} E (r) \, \ mathrm {d} r = \ int \ limits _ {0} ^ {R} \ omega \ cdot r \ cdot B \, \ mathrm {d} r = \ omega B \ cdot {\ frac {1} {2}} R ^ {2}}

It is clear that the voltage that occurs cannot be explained using Maxwell's second equation . Because no matter where the (resting) observer may go with his measuring device: As soon as he has reached his resting state, he always measures a constant magnetic flux density ! From his point of view there are no eddies of the electric field, which is synonymous with the fact that there is no induction. ${\ displaystyle {\ text {red}} {\ vec {E}} = - {\ dot {\ vec {B}}}}$ ${\ displaystyle {\ dot {\ vec {B}}} = 0}$

Calculation with the flow rule

With the flow rule the derivation takes place without integral calculus:

{\ displaystyle U = {\ frac {\ mathrm {d} \ Phi (t)} {\ mathrm {d} t}} = {\ frac {\ mathrm {d}} {\ mathrm {d} t}} ( A (t) \ cdot B (t))}

{\ displaystyle U = {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ left ({\ frac {\ alpha} {2 \ pi}} \, \ pi R ^ {2} \ cdot B \ right) = {\ frac {1} {2}} \, {\ frac {\ mathrm {d} \ alpha} {\ mathrm {d} t}} \, R ^ {2} \ B = { \ frac {1} {2}} \ omega R ^ {2} B}

Here is the angle (in radians) of the circular sector of the area A, which is penetrated by the magnetic field. d / dt symbolizes the time derivative. The sign in has been omitted, the polarity results from the three-finger rule . ${\ displaystyle \ alpha}$ ${\ displaystyle U}$

Generation of alternating voltages

The particular advantage of a generator based on unipolar induction is that it can generate a DC voltage without using a rectifier. Nevertheless, it is also possible to generate alternating voltages with the help of unipolar induction. In the case of the ladder bar moving on rails, for example, the ladder bar can be periodically moved back and forth around an average value or, in the case of the Faraday disk, it can be driven in alternating directions so that the disk rotates sometimes in one direction and sometimes in the other.

Induction law and unipolar induction

Incorrect application of the law of induction can lead to problems in understanding the causes of unipolar induction in the context of classical electrodynamics . This fact is expressed in the Faraday paradox or in the Hering paradox and is partly caused historically by the formation of the term. It is essential for the correct application of the law of induction that the imaginary line along which the induced circulating voltage is to be determined and the electric field prevailing on it are observed from the same reference system. The correct application of the law of induction is possible within the framework of relativistic electrodynamics , a branch of the special theory of relativity , and requires the use of the Lorentz transformation .

literature

On the short theory of the unipolar machine, L. Kneissler-Maixdorf, electrical engineering and machine technology, 61st year, October 1, 1943, issue 39/40, pages 479–486
Unipolar machine with contact rollers for taking off the current, patent specification No. 704671, inventor: Paul Gebhart, patented March 24, 1938
Unipolar machine for small voltages and high currents, M. Zorn, Elektrotechnische Zeitschrift, Volume 61, Issue 16, April 18, 1940, pages 358-360
Unipolar Machines, Association of the Magnetic Field, AK Gupta, American Journal of Physics 31 (1963), p. 428
Unipolar machines, Otto Schulz, 1908, published by Hachmaisler & Thal, Leipzig
Unipolar machine with a deep-frozen field winding, OS 2534511, inventor: Peter Klaudy, applicant: Siemens AG Registration date: August 1, 1975 Int. Cl. H 02 K 31/00
Electrical unipolar machine, OS 2537548, inventor: Dieter Wetzig, applicant: Siemens AG
About unipolar induction, F. Ollendorf, Archiv für Elektrotechnik, XLIV volume, issue 2, 1959

Web links

Answer # 218 ( Memento of November 10, 2012 in the Internet Archive )