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Reduction of Rf Power Loss Caused by Skin Effect Y THP43 Proceedings of LINAC 2004, Lübeck, Germany REDUCTION OF RF POWER LOSS CAUSED BY SKIN EFFECT Y. Iwashita, ICR, Kyoto Univ., Kyoto, JAPAN Abstract Skin effect on a metal foil that is thinner than a skin depth is investigated starting from general derivation of skin depth on a bulk conductor. The reduction of the Figure 1: Half of the space is filled by conductor. power loss due to the skin effect with multi-layered conductors is reported and discussed. A simple application on a dielectric cavity is presented. INTRODUCTION RF current flows on a metal surface with only very thin skin depth, which decreases with RF frequency. Thus the surface resistance increases with the frequency. Because the skin depth also decreases when the metal conductivity increases, the improvement of the conductivity does not contribute much; it is only an inverse proportion to the square root of the conductivity. Recently, it is shown Figure 2: Left: j as a function of x/$. Right: polar plot. that such a power loss can be reduced on a dielectric Markers are put in every unit. cavity with thin conductor layers on the surface, where Although it is necessary to increase the conductance for the layers are thinner than the skin depth [1]. The thin lower RF power loss, the higher conductance leads to foil case is analyzed after a review of the well-known higher current density and thus the thinner skin depth theory of the skin effect. Then an application to a reduces the improvement by the good conductance. It dielectric cavity with TM0 mode is discussed [2]. may be possible to cure this situation if we can control the current density independent of the conductance. SKIN DEPTH OF BULK MATERIAL 7 Typical value of ! is 5.8x10 [S] for copper metal and In materials with conductance ! (>>i"#), Ampere’s that of "# is 1 in vacuum at 18 GHz. Reminding the law becomes: Eq.1, if low loss dielectric material with relative permittivity of more than 107 at such a high frequency "#H = ()$ + i%& E. (1) were available, the RF power loss could be determined by Suppose a system as shown in Fig. 1, Eq.1 finally the dielectric loss other than the conductor loss. becomes: 2 Composite material such as seen in multilayered "x j = i#µ$j, j = $Ey .(2)capacitors may have such characteristics. This will be Under the condition of ! >>i"#, the solution of Eq.2 is: investigated in future. "(1+i)x /# ,(3) j(x) = j0e , # = 2 $µ% SKIN EFFECT ON THIN FOIL where $ is the skin depth. Considering that "#E has Let us consider a case that the thickness of the only one nonzero component of "xEy = "x j # , the conductor is thinner than the skin depth $ (see Fig. 3) and magnetic field in the conductor is derived from Faraday’s the both sides have electromagnetic fields with different law "xEy = #i$µH z as amplitudes. The solution of Eq.2 becomes: " E 2 j(x) = H (0) j e"(1+i)x /# + j e"(1+i)($# "x)/# x y "x j & "x j & z ()f b ,(8) H z (x) = = = = ()1# i j(x) #i$µ #i%$µ #2i 2 .(4) (1 + i)e(1+i)" ()e(1+i)" #$ (1 + i)e(1+i)" ()$e(1+i)" #1 Thus j(x) is expressed by the magnetic field on the j f = , jb = . % e2(1+i)" #1 % e2(1+i)" #1 conductor surface Hz(0): () () j(x) 1+ i H (0)e#(1+i)x /" Figure 4 shows polar plots for j(x) for the case when the = " z .(5) A typical value of $ in copper at the frequency of 3GHz is magnetic fields of both sides have the same amplitude( =1). They are symmetrical with respect to 1µm. Figure 2 shows the current j as a function of x/$. % the origin because the currents from both sides cancel By integrating j, we obtain total current in the conductor: " each other. The linear behaviour can be seen when the J = jdx= H z (0) .(6) #0 The power loss in the conductor can be calculated as # 2 2 H z (0) &µ 2 Pbulk = j " dx = = H z (0) $0 "% 2" ,(7) where 1/!$ is often written as surface resistance Rs. Figure 3: The thickness is thinner than the skin depth. 700 Technology, Components, Subsystems RF Power, Pulsed Power, Components Proceedings of LINAC 2004, Lübeck, Germany THP43 1.5 !=1, "=8 !=1, "=4 1.0 !=1, "=2 !=1, "=1.5 !=1, "=1 0.5 !=1, "=0.5 Im( j ) 0.0 -0.5 Figure 5: Two conductor layers are immersed in -1.0 stepped RF fields: 100%, 50%, and 0%. -1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 1.0 Re( j ) B=1,0.5 !=1 B=0.5-0.0 !=1 Figure 4: Polar plots of complex current densities. The B=1.0-0.426 !=1 0.5 B=0.426-0.0 !=1 markers are at centers of every 1/5 layers in depth: x=0.1!, B=1.0-0.53 !=0.785 B=0.53-0.0 !=4 0.3!, … 0.9!. (currents are normalized by H0/!). Im( j ) 0.0 thickness is less than the skin depth (markers are at the centers of even intervals). -0.5 Let us consider a simple example as shown in Fig. 5, where two layers of thin foils with thickness of just the -1.0 -1.0 -0.5 0.0 0.5 1.0 skin depth ! are immersed in equally stepped RF fields Re( j ) ("=0.5). The current densities in the foils are shown in Figure 6: Polar plots of pairs of complex currents in Fig. 5. Fig. 6. The total current in each layer is just the magnetic field difference between the front and backsides: the currents of both the layers are the same. The total power loss is calculated by the similar way as Eq.7: # 2 2 P2 = j1 + j2 " dx ,(9) $0 () which is 70% of the bulk case. When we optimize the thickness of the first layer # and the current ratio ", the Figure 7: Relative power losses as functions of thickness, minimum power loss becomes 67.9% at #=0.826 and (normalized by the bulk case) "=0.498. Further, when the thickness of the last layer (second layer in this case) is thick enough compared to the skin depth (#=4 in this case), the minimum power loss becomes 65.6% at #=0.785 and "=0.534. When n-layers of equal thickness are immersed in n- equally-stepped RF magnetic fields, the power loss can be given by: n d 2 Figure 8: Minimum power loss as a function of the Pn = ji " dx $ #0 .(10) i=1 number of layers. Such a configuration should be more practical than a 100 configuration with non-uniform thicknesses. Figure 7 10-1 shows the relative power losses as functions of the 10-2 normalized thickness #=d/!. The thicknesses that show 10-3 minimum power loss decrease as the numbers of layers 10-4 Transmission 100THz (!=3µm) 10-5 1THz increase. The minimum power loss is shown in Fig. 8 as 10GHz -0.5 100MHz a function of the number of layers. It shows n 10-6 -3 -2 -1 0 1 " dependence, which can be also derived from an expansion 10 10 10 10 10 of the power loss formula. Therefore, the RF power loss Figure 9: Transmission through a thin conductor foil may be reduced by this geometry, if the current on each with thickness of #!. layer is well controlled. The improvement, however, is limited by the absolute thickness of each layer: it should foil. Because the transmission is less than 1% at 10GHz be enough thicker than the inter-atomic distance. On the even if the thickness is one-hundredth of the skin depth, other hand, the power loss can be more than the bulk case, the two regions, the front side and the backside, are which may be useful for absorber applications such as isolated by the foil: the left hand side field can hardly EMI shields. penetrate the foil and no power can be delivered to the Suppose a plane wave comes from the left in Fig. 3. other side. A plane wave with an angle other than The transmission T through the foil is shown in Fig. 9 as a normal makes a different situation from this and a function of #., where #! is the thickness of the copper solution to distribute the current to each layer is reported Technology, Components, Subsystems 701 RF Power, Pulsed Power, Components THP43 Proceedings of LINAC 2004, Lübeck, Germany Spacer in Ref[1]. This method, however, requires narrow choice R in the material inserted between the conductor layers. Dielectric material r1 r2 r Conductor DIELECTRIC CAVITY WITH THIN FOIL Let us consider a flat dielectric cylindrical cavity with Figure 11: Dielectric cylindrical cavity with extra electrodes that have washer shape. The thicknesses are TM010 mode as shown in Figure 10. Suppose a high exaggerated: inside of the each conventional disc dielectric constant !r, the boundary condition at the perimeter (r=R) is almost magnetic, where the magnetic electrode, one smaller washer shape electrode is located field is normal to or zero on the boundary. The electric with a small distance from the disc. field Ez can be described by the Bessel function as: Figure 12 shows an axisymmetric simulation code Ez (r) = Ez (0) J 0 (kr), k = x1' R,(11) result of the Q enhancement as a function of r1/R.
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