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I NSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 25 (2004) 463–468 PII: S0143-0807(04)75775-X
Experiments with eddy currents:
the eddy current brake
Manuel I Gonz ´alez
Departamento de F´ısica, Universidad de Burgos, 09006 Burgos, Spain
E-mail: [email protected]
Received 6 February 2004 Published 20 April 2004 Online at stacks.iop.org/EJP/25/463 ( DOI: 10.1088/0143-0807/25/4/001 )
Abstract A moderate-cost experimental setup is presented to help students to understand some qualitative and quantitative aspects of eddy currents. The setup operates like an eddy current brake, a device commonly used in heavy vehicles to dissipate kinetic energy by generating eddy currents. A set of simple experiments is proposed to measure eddy current losses and to relate them to various relevant parameters. Typical results for each of the experiments are presented, and comparisons with theoretical predictions are included. The experiments, which are devoted to first-year undergraduate students, deal also with other pedagogically relevant topics in electricity and magnetism, such as basic laws, electrical measurement techniques, the sources of the magnetic field and others.
1. Introduction
Eddy currents are one of the most outstanding of electromagnetic induction phenomena. They appear in many technical problems and in a variety of everyday life situations. Sometimes they are undesirable because of their dissipative nature (e.g. transformer cores, metallic parts of generators and motors etc). In many other cases, however, eddy currents are valuable [1] (metal detectors, coin recognition systems in vending machines, electricity meters, induction ovens, etc). However, little attention is paid to eddy currents in many of the textbooks commonly used in introductory physics courses [1–3]: they are often dealt with only from a phenomenological point of view, and they are considered in some cases only as a topic for optional reading [1, 3]. Furthermore, most of the commercially available experimental setups concerning eddy currents treat only their qualitative aspects. This paper presents a set of laboratory experiments intended to help students better understand the phenomenon from a quantitative point of view. The experiments may be helpful to physics and electromagnetism students in first-year undergraduate courses.
0143-0807/04/040463+06$30.00 ©c 2004 IOP Publishing Ltd Printed in the UK 463
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ω
Figure 1. A sketch of the eddy currents in a rotating disc. The crosses represent a steady magnetic field perpendicular to the plane of the disc. According to Faraday’s law, eddy currents appear in those points of the disc where the magnetic field increases or decreases.
2. Theoretical foundation
Induced currents appear when electrical conductors undergo conditions of variable magnetic flux. In particular, we talk about eddy currents when bulk conductor pieces instead of wires are involved. There are two basic procedures to achieve such conditions:
- exerting a time-varying magnetic field on a static piece;
- exerting a steady magnetic field on a moving one. An example of the latter class will be investigated. It consists of a rotating metallic disc, which is subjected to the magnetic field present at the gap of an electromagnet. Eddy currents appear inside the disc and brake its rotation. This is the foundation of the electromagnetic braking systems used by heavy vehicles such as trains, buses or lorries [2]. Even in such a geometrically simple case, the pattern of eddy currents is complex. Figure 1 and [3] show simplified sketches of this pattern. It is easy, however, to obtain an approximate expression for the power dissipated by eddy currents. Since the magnetic field B is steady, the induced electric field in each point of the disc is given by E = v × B, where v is the velocity of that point [4]. Instead of measuring B directly, we will relate it to the excitation current Iex in the coil of the electromagnet, which is easily measurable. For the moment we will assume that B is proportional to Iex (the validity of this hypothesis, which is not true for magnetic media, will be discussed later). Then the following proportionality law holds:
E ∝ ωIex
where ω is the angular speed of the disc. This means that for any loop of eddy current the induced electromotive force, being the line integral of the induced field, is also proportional to ωIex. Finally, the basic laws of electric current state that the power dissipated in that particular loop is proportional to the square of the electromotive force and to the inverse of the electrical resistivity of the disc. The same holds for the power dissipated in the whole disc:
Pe = K
ω^2 I (^) ex^2 ρ
466 M I Gonz ´alez
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0.0 1.0 2.0 3.0 4.0 5. Iex (A)
t (s)
Figure 3. Braking time for the copper disc versus excitation intensity. The initial speed of the disc was set to 200 rpm.
must keep in mind that the larger the excitation intensity is selected, the larger the voltage applied to the motor must be to achieve that initial speed. Figure 3 shows how larger excitation intensities make the braking time shorter as a result of more powerful eddy currents. Error bars have been set to 0.3 s, a typical uncertainty when using stopwatches. Equation (1) implies that the eddy current braking torque is proportional to the instantaneous angular speed. However, the results plotted in figure 3 are not suitable for verifying this fact, due to the lack of a known model for the internal braking torque acting on the motor. Therefore, the result of this first experiment cannot be numerically tested.
4.2. Eddy current losses versus angular velocity
To test the proportionality between Pe and ω^2 shown by equation (1) a fixed excitation intensity Iex must be chosen. Then the voltage supplied by the power source of the motor must be varied in order to select various angular speeds. For any chosen speed ω of the disc the power consumption of the motor Pm (Iex , ω) can be calculated as the product of its voltage and intensity. The power consumption Pm ( 0 , ω) when the electromagnet is turned off must be computed in the same way. Then the power dissipated only by eddy currents is simply:
Pe (ω) = Pm (Iex , ω) − Pm ( 0 , ω) (Iex fixed). (2) Results are plotted in figure 4. Standard error analysis techniques have been used to calculate error bars, assuming uncertainties of 0.2 V for voltages, and 1 mA for intensities. As we can see, the constancy of Pe (ω)/ω^2 within our experimental uncertainties is consistent with equation (1).
4.3. Eddy current losses versus excitation intensity
Equation (1) states that Pe is proportional to the square of the excitation intensity in the coil. To verify this result a fixed angular speed ω must be chosen. Then the power consumption of the motor for various excitation intensities as well as for Iex = 0 must be measured. The eddy current losses are calculated in the same way as given by (2)
Pe (Iex ) = Pm (Iex , ω) − Pm ( 0 , ω) (ω fixed). (3)
Experiments with eddy currents: the eddy current brake 467
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50 100 150 200 250 ω (rpm)
Pe
/ ω
2 (mW/rpm
2 )
Figure 4. An example to test the proportionality between Pe in the copper disc and ω^2. The excitation intensity was fixed to 3.0 A.
0
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1.0 2.0 3.0 4.0 5.0 6. Iex (A)
P
/Ie ex
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Figure 5. An example to test the proportionality between Pe in the copper disc and I (^) ex^2. The angular speed was fixed to 200 rpm.
Figure 5 shows a set of typical results. It appears that the proportionality between Pe and I (^) ex^2 does not hold, especially at low intensities. Nevertheless, we must recall that the core of the electromagnet is made of iron, i.e. a nonlinear medium, while equation (1) was obtained under the assumption of proportionality between B and Iex. In fact, it would be interesting to use a coil alone as a truly linear source of magnetic field. Unfortunately, the absence of a ferromagnetic core results in low magnetic fields and therefore Pe becomes too low to be measured with conventional ammeters and voltmeters.
4.4. The influence of disc resistivity
Finally the inverse proportionality between Pe and ρ expressed by equation (1) can be tested. To carry out such a test, both rotation speed and excitation intensity must be fixed, and the power consumed by the motor with each of the discs attached to it must be calculated. Pe is
