Shielding of Magnetic Fields by Eddy Currents

1 Payne : Eddy current Shielding 1 Shielding OF Magnetic Fields BY EDDY Currents Alan Payne 2016 Alan Payne asserts the right to be recognized as the author of this work. Enquiries to Payne : Eddy current Shielding 2 TABLE OF CONTENTS 1. INTRODUCTION .. 3 2. Magnetic Fields .. 3 3. THEORY FOR Magnetic Shielding BY A SINGLE PLATE .. 4 4. THEORY FOR Shielding BY A 7 5. COMPARISON WITH PUBLISHED BOX MEASUREMENTS .. 9 6. COMPARISON WITH PLATE MEASUREMNTS .. 10 7. HIGH PERMEABILITY MATERIALS .. 11 8. MESH SHIELDS .. 17 9. PERPENDICULAR FLUX .. 19 10. CONCLUSION .. 20 APPENDIX 1 : MEASUREMENT OF ATTENUATION THROUGH METAL SHEETS .. 21 APPENDIX 2 : DETERMINATION OF DILUTED PERMEABILITY .. 21 APPENDIX 3 : BOX RESISTANCE .. 23 Payne : Eddy current Shielding 3 Shielding Magnetic Fields BY EDDY Currents It is known that thin metal films can provide very good Shielding of alternating Magnetic Fields , even at low frequencies.

2 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. 1. INTRODUCTION The Shielding of alternating Magnetic Fields is often assumed to require a metal with a high permeability such as iron, or require a conductor which is very thick such that it exceeds the skin depth in the metal. However it has been shown that a thin aluminum foil can provide a high level of attenuation at frequencies where it is much thinner than a skin depth (ref 1 Weston). The explanation for this is that eddy Currents are induced in the conductor by the incident field and these produce a Magnetic field which opposes the applied field . No accurate theory has been presented to date for the cancellation which eddy Currents can produce, and such a theory is given here.

3 This gives the Shielding provided by a single plate and this is then extended to enclosures. 2. Magnetic Fields Magnetic Fields and Electromagnetic Fields There can be some confusion between a Magnetic field and an electromagnetic field , and the difference is outlined as follows. When a direct current (dc) is passed through a wire there is a Magnetic field around the wire and an electric field between its ends. If the current is alternating these Fields are still present and they also alternate, but a new field is generated because the electrons in the wire are now accelerating. This new field is called an Electro- Magnetic field (EM). Close to the wire this field is very weak compared with the Magnetic and electric Fields , but as the distance from the wire increases the Magnetic and electric Fields decrease at a higher rate than the EM field . They are equal in amplitude at a distance of about 1/6th wavelength, and beyond this distance the radiated field dominates.

4 For instance at 1 MHz the Fields are equal at a distance of about 50 meters. In this article it is only the alternating Magnetic field which is being considered. The Electric field around a Magnetic field When a loop of wire encloses a changing Magnetic field an emf can be measured at the open terminals of the loop. This effect is well known and is the basis of transformer action, but what is less well known is that the emf exists whether the wire is present or not. Around every Magnetic field there is an electric field , and the wire is a device for measuring this. So when a Magnetic field is incident upon a metal plate, an electric field is induced in the plate and this drives a current whose magnitude is limited by the resistance of the conducting path. The magnitude of the electric field is given by Lenz s law, which states that the induced emf is equal to the rate of change of Magnetic flux e = - d r / dt volts. Payne : Eddy current Shielding 4 3.

5 THEORY FOR Magnetic Shielding BY A SINGLE PLATE Introduction When an alternating Magnetic field is perpendicular to a conducting plate, eddy Currents are generated in the plate as shown below : Figure Circulating Currents and Resultant current Here the Magnetic flux is represented by 25 discrete flux concentrations and around each one there is an electric field which causes a circular current . The direction of these Currents is such that they produce a Magnetic field which opposes the applied field . Over most of the plate these Currents cancel to produce a resultant current around the edge, which opposes the applied field . A very good video demonstration of this is given in ref 2, where an iron plate is placed inside an induction coil and the edges of the plate become red hot while the rest of the plate remains black. (NB the audio explanation with this video says that the effect is due to skin effect, rather than eddy Currents ).

6 In contrast to the above, this paper considers flux which is tangential to the surface, as shown below : Figure Tangential Magnetic flux The above shows a single plate, and the analysis is extended to a box structure in Section 4. Surprisingly this tangential analysis gives very good agreement with published measurements, even when these appear to be due to a perpendicular field . This is discussed more in Section 8. Payne : Eddy current Shielding 5 Theoretical Analysis The configuration to be analysed is shown below. A plate of length p, width wp, and thickness t is illuminated by a tangential Magnetic flux. Figure Analysed configuraton The flux is shown entering the edge of the plate on the side designated as the width. Note that the width of the plate is not necessarily the shortest dimension but is defined here as the edge into which the flux is directed. The length of the plate is in the direction of the flux path through the plate.

7 This flux induces an emf in the plate and this leads to a circulating eddy current , the magnitude of which is determined by the resistance of the path which the current follows. In turn this eddy current generates a Magnetic field which opposes the incident field , and partially cancels it. The resultant field is therefore lower than the incident field and it is assumed that it is this resultant field which appears on the other side of the plate. The resultant flux density Br is given by : Br = Bo-Be where Bo is the incident flux density Be is the flux produced by the eddy current The first thing to be determined here is the flux produced by the eddy current Be. As shown in the Figure the cross-section of this current flow is very long and narrow, and it extends down the whole length of the plate p. There are therefore two parallel current sheets carrying current in opposite directions and it is assumed here that the flux density is the same as that from two parallel wires.

8 (This cannot be justified on purely theoretical grounds but it does lead to an equation which agrees extremely well with published measurements). It is seen from Figure that the two current sheets are spaced by a distance somewhat less than the thickness t, and let this be t . The flux density is then assumed to be : Be= o rm i / (2 t ) The relative permeability rm is unity for most metals, such as copper and aluminium. Permeable metals such as iron and steel are considered in Section 7. Payne : Eddy current Shielding 6 The emf e induced in the loop is due to the resultant flux r : e = d r / dt which for sinusoidal excitation, and area of loop wp t will be : e = j r = j wp t Br For a loop resistance R, the current i induced in the loop will be i= e/R : i = j wp t Br / R Substituting this into Equation : Be= j o rm 2 f wp t Br / R/(2 t ) = j o rm f wp Br / R Notice that flux due to the eddy current Be is independent of t.

9 Normalising to incident flux density Bo, gives Be (= Be / Bo), and similarly Br , and then Equation becomes : Be = j o rm f wp Br / R From Equation and normalising to Bo : Br = 1 - Be Br = 1 - j [ o rm f wp Br / R] Br + j [ o rm f wp Br / R] =1 Br = 1/ [1+ j o rm f wp / R] The resistance R is equal to /A. Notice that the width and length of the plate are defined with respect to the flux, and for the current these are reversed and so R is equal to 2 wp /( p t/k), where t/k is the width of the conducting area. Equation then becomes : Br / Bo= 1/ [1+ j ( o rm f p (t/k) / (2 ) ] The phase angle is given by : = tan -1 (- o rm f p (t/k) / (2 ) radians (k shown later to be equal to 4) [this comes from y=1/(1+jk) = (1-jk)/[(1-jk)(1+jk)]= (1-jk)/ (1+k2). The angle of this is tan-1 (-k/1)] It is conventional to express the Shielding performance as the Shielding effectiveness (SE) and this is the inverse of Equation Taking its modulus gives : |SE|plate = [1+ {( o rm f p (t/k) / (2 )} 2 ] Using this equation good agreement with experiment and with published measurements was obtained with k=4, so the average conducting width is of the conductor thickness.)))

10 So we can imagine that the current is Payne : Eddy current Shielding 7 constant at its maximum value to a depth of t/4 from each surface, and zero at other depths. More likely it could change linearly from one surface to the other as shown below: Figure current in Conductor Cross-section With k=4 Equation becomes : |SE|plate = [1+ {( o rm f p t / (k1 )} 2 ] where o = 4 10-7 H/m rm is the relative permeability (normally unity) f is the frequency in Hz wp is the width of the plate in m t is the thickness of the plate in m is the resistivity of the conductor p is the length of the plate in m k1 =8 for a plate (see later for a box) When the frequency is high enough that the factor {( o rm f p t / (k1 )} is much greater than unity Equation approximates to : |SE| ( o rm f p t / (k1 ) This shows that at high frequencies the Shielding effectiveness is proportional to frequency, so the attenuation through the conductor increases at 20 dB per decade of frequency.)))