OPTIMAL DIMENSIONING OF COMPONENTS FOR A HIGH VOLTAGE FEEDTHROUGH Marjan Jenko\ Anton Mavretič^ ^ Laboratory for Electrical Engineering and Digital Systems, College of Mechanical Engineering, University of Ljubljana, SLovenia ^Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, USA Key words: components for power delivery, dielectric materials, dielectric strength, dissipation constant, feedttirough, high voltage applications, high voltage breakdown, RF power, RF power equipment, unbalanced load impedance Abstract: The design of high voltage feedthrough components needs to satisfy mutually contradictory requirements. Selected geometries, dimensions, and materials of a high voltage feedthrough need to prevent voltage breakdown under worst case conditions, while size and weight are often constrained. The compact size requirement is less important in massive systems (distribution of electric energy), but it becomes critical with requirements of limited space and/or low mass. Both constraints, on size and weight, are crucial in RF power delivery systems for plasma processing in the semiconductor industry and in satellite-mounted instruments for space exploration, such as instruments for solar wind measurements. Expressions for optimal dimensioning of a high voltage feedthrough are derived in this paper for the case of delivering RF energy to a plasma chamber via an impedance matching network. Derived geometries and expressions are useful in design of high voltage feedthrougs in RF and other engineering areas (instruments for space exploration, instruments for high-energy pliysics experiments, X ray systems, systems for distribution of electric energy). Določitev optimalne geometrije in izpeljava izrazov za dimenzioniranje komponent visokonapetostnega prehoda Kjučne besede: dielektrični materiali, dimenzijska optimizacija, močnostne komponente, RF energija, RF močnostna oprema, neuglašena impedanca bremena, visokonapetostne komponente, visokonapetostni preboj Izvleček: Konstruiranje komponent visokonapetostnih prehodov mora zadovoljiti izključujoče se zahteve. Velikosti in materiali komponent visokonapetostnega prehoda morajo preprečiti napetostni preboj pri najneugodnejših pogojih delovanja, in sistemske zahteve pogosto omejujejo velikost in maso visokonapetostnih komponent. Zahteva po majhnih dimenzijah in masah visokonapetostnih prehodov je manj pomembna v velikih sistemih, kot na primer pri distribuciji električne energije. Zahteva po majhnosti pa je kritična, kadar so dimenzije in masa celotne naprave vnaprej omejene. Kompaktne izmere in majhna masa visokonapetostnih sistemov sta elementarni zahtevi v a) konstrukciji sistemov za generiranje in dovajanje elektromagnetne energije v radijskem frekvenčnem (RF) območju za vzbujanje RF plazme za prizvodne procese v izdelavi mikroelektronskih vezij, in b) v konstrukciji satelitskih sistemov in instrumentov. Vtem prispevku so določene optimalne geometrije in so izvedeni izrazi, potrebni za optimalno konstruiranje visokonapetostnega prehoda pri dovajanju RF energije v plazemsko komoro preko sistema za impedančno prilagajanje. Izvedene geometrije in izrazi so uporabni za načrtovanje visokonapetostnih prehodov v RF tehniki in na drugih področjih (instrumenti za raziskovanje vesolja, instrumenti za eksperimentalno delovvisoko-energetski fiziki, roentgen-ski sistemi, distribucija električne energije). 1. Introduction A universal problem in the design of RF power equipment is the transfer of RF power across equipment walls. Typical examples are energy transfer out of RF generators, into and out of matching networks, and into loads. At high powers, the physical design of such feedthroughs runs into the difficulty of satisfying mutually contradictory requirements. One requirement, related to voltage, is that there be sufficient separation between the center conductor and the wall. The other requirement, not fundamental but quite common, is that of over-all size reduction - which may limit the space available for the feedthrough. When the impedance is controlled, as out of a generator and into a matching network (the two being connected by a 50 Ohm ca- ble), tested commercial solutions exist for different power ranges. It is at the interface of the loads (e.g., the RF plasma chamber) and matching networks /1,4/, that the RF and mechanical designers face the challenge of geometric optimization /3/. What makes this interface critical is the uncontrolled impedance of the load /2, 5/. For a given power, whose maximal value is known as the power rating of the generator Pg,„ and for variable load impedance, the voltages that may appear at the feedthrough can reach extremely high values. Hence, the feedthrough design voltages are not those expected in steady state operation, but those that may be generated by the worst transients, however short. Thus, for a generator power rating and maximal transient load impedance is V ^ p ^ eJUoad gen_rated [y j max and the feedthrough design voltage (peak value) is F ,, = ^ J2P 7 ■ gen_ rated max (1) where K is the designer's safety factor. Fortunately, the feedthrough design problem naturally splits into two parts. One is the optimization of conductor shapes, regardless of the required maximal voltage rating; the other is the selection of its overall dimension. The first part, which is pure physics, can be solved once for all for any type of feedthrough geometry - which is done in the present paper for an easily manufactured feedthrough. The second part depends on the application. Here, we only suggest guidelines. 2. Derivation of expressions for optimization of shapes and dimensions of conductors We state the problem as follows: For a given round hole opening of diameter D in a cabinet wail determine the feedthrough shape that withstands the highest RF voltage if the dielectric is air Here, D is the variable that characterizes the physical size of the solution (second part of the problem). As we shall see, it is proportional to Vmax. The mechanical feedthrough model is that of a cylindrical conductor (usually made of copper tubing) of diameter d passing at a right angle through the center of the hole. The support issue is irrelevant in principle, provided the corona paths along insulator surfaces are sufficiently long. The voltage limit for this model is defined by the onset of arcing, which would take place between the center conductor and the wall over a distance (D - d)/2. As arcing begins at local ionization spots^ when the dielectric field exceeds the medium's characteristic breakdown value, (about 1000 V/mm for dry air), the often quoted computation of electric fields as voltage divided by distance regularly leads to grossly under-designed feedthroughs. This is because the voltage over distance expression is valid only between parallel capacitor plates. In all other cases, we must use the exact definition of the electric field, which is the gradient of the potential function. This is particularly true near all edges. The edge of the window in a cabinet wall is the critical one for feedthroughs. It thus follows that all radii of curvature must be maximized, not only the distances between conductors. This is why we install into the window a tube of outer diameter D and wall thickness w, as shown in Figure 1. The length of this tube is not important. It can be optimized with respect to other considerations (e.g., mechanical structure). The optimal wall thickness w of the tube will be determined below. ' A'/ / / /y/yyTX/j^'/'^r/yAr //''//■////'' / / / / / / Figure 1 Tube-type high voltage feedthrough. Disregarding for the time being the edge effects at the end of this tube, our first task is to determine the optimal diameter of the inner conductor in terms of hole opening of diameter D in a cabinet wall. To this end, we view the geometry as a cylindrical capacitor, whose inner electrode is of outer radius a = d/2 and outer electrode of inner radius b = (D- 2w)/2. Solving the Laplace equation for the space within the capacitor, i.e., V^0 = 0, the potential function 0=0 (r) between the two conductors is F / L \ -In In a V / b r V y The magnitude of the electric field is then £ = | V4>|, i.e.. E = dr -V rln / \ a J , V y Clearly, its maximal value is at the minimum value of r, which is r = a: f =. max -F a In / \ a 1 (2) This value is to be minimized by properiy selecting the ratio a/b. For a given b, we thus require dE^^^^Jda =0. 1 if the medium is not air, but a dielectric, the language may change from "arcing" to "punch through", but the arguments and the calculations are identical. dE„ V da aHn b / b n In + a- b i a b V V J / = 0, which implies In / \ a J> , V y + 1=0. Hence, the solution is a = hje. This establishes one relationship between D, d, and w: , D-2w a -- . e The next task is to determine the optimal value of w. The following intuitive arguments give us a starting point: To minimize the field inside the capacitor, it is better to have a small value of w, as this leaves a greater spacing between the conductors. A thin wall, however, implies a small radius of curvature at the edge, namely w/2, as it is shown in case C in Figure 2. Hence, from the point of view of edge effects, it is better to have a thicker wall. \D HV coiuiiictor i force iuics \D H V conductor lyrce lujes / fcccitliroiigh end, vv is about zero feedthrougli end, cut feedtlirough end, rounded cylinder approximation of force lines ca.sc C Figure 2 Ends of tube-type high voltage feedthroughs. An exact solution to the problem of optimal wall thickness would also give us the optimal shape according to which the edge should be formed. Such a shape is not present in cases A and B in Figure 2 that are simple to manufacture. From the standpoint of machining, one would easily decide for either case /4 or case B in Figure 2. Figure 2, case C is quite close to an exact solution. Additional computational and machining efforts of an exact solution of the optimal edge shape would hardly be justified by the marginal improvement. We thus assume that the edges terminate in a semicircular profile of radius w/2. For a heuristic estimation of the minimal value of w/2, we note that the density of force lines near the edge (but inside the cylinder) is equal to the density at the inner conductor divided by e. Hence, in a good approximation, we may increase the density of lines at the edge of the cylinder by a factor of e without degrading the voltage rating of the device, as suggest the geometries and force lines shown in dashed lines in Figure 2, case C. This occurs if we take the radius of curvature at the edge to be the radius of the inner conductor divided by e, i.e., w = dje. With the previous equation, this gives us the complete solution: D w = ' d = ew. Numerically, W-- d = D 9.39 D (3) (4) 3.45 The next task is to relate the window's diameter D to the absolute maximum of peak RF voltage, Vmax, (1) that could possibly appear on the conductor. The governing equation is (2). Combining it with relation (1), we get F = max -K-n^x _ ^i'^Pgen..rated a In b V y flln a V y Substitution of a/b = 1/e and d D-lw 2 2e leie'+l) D yields E.. e^D Expressing power in kW, all dimensions in meters, and taking Emax = 10®V/m, we finally get D^K \l2P. lOOOe generated 7 max Numerically, D cm ^0JKj2P kW' or D cm ■ generated The safety factor K remains to be selected. For a 50 percent safety margin, for example, we would have K = 1.5, and hence, the simple rule of thumb Z)rcm1 = F TkV (5) if Vmax has already been estimated with a reasonable safety margin, one can take K = 1, which yields D cm = 0.7K_ TkV (6) Thus, at least one centimeter of window diameter opening is required for every seven hundred volts of peak RF voltage. This assumes that the edge of the window is mechanically terminated with a tube that leads through the window in the equipment wall. Figure 1. The tube is to be made according to the optimal dimensions derived above. Any other dimensioning makes matters worse. 3. Worked out examples Let us assume that the RMS voltage at full generator power in a matched condition is 2 kV. The peak voltage is then which can be conservatively rounded to 3 kV. Let us further assume that experience and computations indicate that in the case of worst transient mismatch the steady state voltage gets doubled, at most. Then, we may take Vmax = 6 kV. Equations (6), (4), and (3) then yield D = 0.7*6 = 42 cm, 4 2 t/= — = 1.24 cm, 3.4 HV condiiclür :U . c/ f iV coiuKictor Uj \ d 1 [V conductor ü.d cqiiipmem wall, u- is about zero cylinder approxiiiration of force lines equipment wail, cut equipment wail, rounded case B case C Figure 3 High voltage feedtiirough without a buiit-in tube. These results are for framed feedthroughs, which are often a natural mechanical solution regardless of voltage rating considerations. If the frame is not needed, however, its length may be reduced to the minimum, which is its wall thickness, w. It may then be implemented as an integral part of the housing wall, provided its edges are machined to a semi-circular profile, as illustrated in Figure 3, case C. The required opening D is then smaller, since it does not have to accommodate the frame. Geometries and force lines shown dashed in Figure 3 are identical to those in Figure 2, and so is the analysis. Results, i.e., modified design equations are: D D cm = 0.6^ F TkV (7) (8) (9) Cases A and B in Figure 3 would be simple to manufacture, but such geometries are inadequate for optimization of feedthrough dimensioning, since sharp edges imply intense local peaks in force line densities. 4 2 w =-= 0.45 cm. 9.4 Hence, the smallest possible robust feedthrough requires a window opening of 4.2 centimeters, a center conductor 1.24 cm in diameter and a window frame of wall thickness 0.45 cm. Figure 1 shows the general appearance. Both ends of the tube are rounded as shown in Figure 2, case C. The length of the tube is arbitrary, and may be very short. For a straight feedthrough, case C in Figure 3 (no separate tube), equations (9), (7) and (8) yield the solution = 0.6*6 = 3.6 cm,