Magnetic Field and Inductance Calculations in Theta-Pinch and Z-Pinch Geometries

Transcription

1 Magnetic Field and Inductance Calculations in Theta-Pinch and Z-Pinch Geometies T.J. Awe, R.E. Siemon, B.S. Baue, S. Fuelling, V. Makhin, Univesity of Nevada, Reno, NV S.C. Hsu, T.P. Intato Los Alamos National Laboatoy, Los Alamos, NM NATIONAL LABORATORY EST. 943

2 Abstact Two codes have been developed at the Univesity of Nevada Reno which model solid metal o wie-wound conductos by assuming aays of thin-wie loops. The fist code applies to the fomation and tanslation of an FRC plasma, whee theta coils ae used both to ceate the plasma, and to geneate tanslation fields. Shielding is equied to suppess lage voltage tansients and potect sensitive components. The esulting field involves diven cuents in the theta coils and eddy cuents in the shielding stuctues. Ou two-dimensional -z code calculates eddy cuent induction, esistive diffusion, and the esultant magnetic field of cylindically symmetic conductos. We use fast and accuate elliptic integal suboutines fom MATLAB to solve fo the time dependent cuent flowing though each loop and the esultant magnetic field configuation. We will show computational esults fo FRX-L hadwae design at Los Alamos National Laboatoy. The second code calculates fields and inductances fo conductos in a z-pinch geomety. The two-dimensional -theta code was witten to help with design of a flux compession expeiment to be done at the Atlas pulsed powe facility. In this expeiment, shunt inductos will divet a potion of the main bank cuent onto a had coe inside the line. The line will then be imploded, compessing the injected flux. The code calculates the shunt inductance, the mutual inductance between the shunt inductos and the had coe, and the esultant cuent division of the system. Numeical esults will be compaed to data obtained fom a small pulsed powe system that dives a pototypical inductive divide assembly.

3 Physics Equations fo RZ Geomety Ou fist code, which we call EDDY, models cylindically symmetic solid metal objects with tightly packed aays of cicula wie loops, each with cicula coss section. We need only know thee geometic paametes of the loop, namely the adius of the loop (b), this distance of the loop cente fom the z=0 plane (z), and coss sectional adius of the loop (a). When calculating the vecto potential, magnetic field, and mutual inductance value fo an isolated loop, we assume the coss sectional adius (a) appoaches zeo, so that these values ae efeenced to a single point, (,z). When calculating the self inductance and esistance of the loop, the finite coss section must also be consideed. We fist find the vecto potential at (, z) due to a loop at (, z), θ θ θ θ π μ π ˆ cos ) ( cos 4 ), ( = z z d I z A In ode to damatically educe the computational time, we utilize fast elliptic integal statements. The vecto potential becomes ( ) ( ) 0 ) ( 4 ˆ ) ( ) ( ), ( z z k whee k k E k K k I z A + + = = θ π μ

4 To calculate the mutual inductance between two loops we elate the field geneated fom the cuent in loop one to the flux passing though loop two. We know Φ = MI = π A (, ) z whee M is the mutual inductance between the two loops, thus we find M = μ 0 ( k ) K ( k ) k E ( k The esistance and self inductance of the loop above ae found to be L b 8b 7 R = ρ = ρ and L = μ 0b A a ln a 4 With this infomation, we may simply use the Kichoff elation which states that shoted conductos will have no change in potential as we tavese the loop. Thus we have di di i j L i + R i I i + M ij = 0 dt dt Hee we have found n coupled diffeential equations that goven the tansient cuents in each loop; this is pecisely the type of poblem that Matlab is suited to solve. We ae now in a position to find the cuent flowing though each wie element as a function of time, allowing fo field calculations, including diffusion into the conductos. j )

5 Test : Diven Solenoid Ou initial test was to dive a KA solenoid with 00 tuns/mete. We expect to see B=0.57 T inside the solenoid, with the field dopping to zeo outside. The uppe plot shows flux contous paallel to the axis and deceasing in magnitude as we decease in adius. This figue shows one poblem with ou simulation method. By using individual wies to simulate solid objects, we have localized cuents athe than a continuous cuent density. Theefoe, we find that nea the loops the field is dominated by the local cuent. This effect can be seen by the finged contou along the suface of the conducto. In the second figue we look at the magnitude of B as a function of adius. Hee the code has matched both qualitative and quantitative expectation

6 x 0-3 Test : Coupling of Two Wies Next we examine the most simplistic case of eddy cuent induction. To the left we see the magnetic flux contous fo a pefect cuent souce loop located at (.5,.0) and a shoted inductively diven loop at (.,.03). The flux contous shown ae fo time t=5e-4 seconds, which is the specified ise time of the dive coil. We see that the eddy cuent induced in the shoted loop has the effect of limiting the flow of flux though the plane of the loop. To the lowe left, we see the sinusoidal dive cuent in blue, along with the induced eddy cuent in the shoted loop shown in geen. As expected, the eddy cuent is of lowe magnitude than the dive cuent, and thee exists a slight phase shift. The figue quantitatively agees with the Kichoff elations peviously stated.

7 Test 3: Magnetic Field Diffusion As the final and most igoous test of code pefomance we use a diven solenoid to ceate a pue B z field, and analyze the diffusion of this field into a coaxial cylindical conducto. The long shield spans adially fom to.8cm. We chose to analyze the axial component of the magnetic field at two adial locations: A=.75cm and B=3cm. At these points, we ae fa enough fom the walls of the cylinde to be able to negate the eos fom the dominant local fields. We fist test code convegence by inceasing the numbe of wie Diffusion of B z into Cylindical Shield 0.8 B z (,t) columns composing the shield fom 0.6 to,3,4, and finally 8; convegence I 0.4 Eddy I was clealy shown. We then checked Dive 0. that the code was conveging to the A B 0 coect solution by compaing with MHRDR, a tusted code. The -0. diffusion fields fom the two codes ae plotted to the ight. When using at least fou columns the codes agee -0.8 vey well, convincing us that EDDY - calculates field diffusion with Time (sec) acceptable accuacy. Axial Magnetic Field (AU) Dive Field MHRDR Column Columns 4 Columns 8 Columns

8 Field Calculation of Fomation/Tanslation Region of Line on Plasma Expeiment 0.0 m 0.06 m Next, we will calculate the eddy cuents and fields in a geomety elevant to a plasma fomation, tanslation, and implosion expeiment. The diagam shows geneic geometic featues of such a system. We include the fast ising, high voltage conical theta coil fo plasma fomation and initial axial acceleation, and the tanslation egion consisting of 6 slowly ising, low voltage, multi-tun coils. These coils poduce a magnetic field which slowly diffuses though the shield and line, ceating magnetic flux inside the line. The spacing of the coils is educed at each end, ceating mio fields which help to tap the plasma in the line egion. We wish to examine seveal featues of this geomety including the diffusion of the fields though the cylindical line and shield, the necessity of pope shielding to potect expeimental hadwae, and the tansient behavio of the fields thoughout the system. Coil 7 A Coil 6 Coil 5 B Coil 4 Coil 3 C Coil Aluminum Line Coil D E F x x x X X X X X X X X X X X X Steel Shield.7 m.7 m.7 m Mio.0508 m.0335 m.006 m Fast ising, high voltage conical theta coil fo plasma fomation and tanslation Slow ising, low voltage multi-tun coils geneate magnetic fields in the tanslation egion

9 Magnetic Field (T) Magnetic Field (T) Slow Field Diffusion Though Cylindes Time (sec) F C E B D A On-Axis Magnetic Field Magnitude Location on z-axis (m ) t=0.6ms t=.ms t=.8ms t=.4ms t=3.0ms Diffusion of Slow Magnetic Field Fist, we analyze the diffusion of the magnetic field fom the slow coils though the cylindical conductos. Each of the six coils consists of 3 evenly spaced wies, and each is diven as a pefect sinusoidal cuent souce with peak amplitude of 0KA. The ise time of the coils is set to be ms. At this fequency, we find that slightly moe than half of the field manages to penetate though the conductos. We also see the effect of the mio coils, as stonge fields exist at locations D and F, than at location E. In the second plot we show the time evolution of the on-axis magnetic field as a function of axis location. We again see the inceased field stength at the mio locations.

10 Effects of Shielding Next, we demonstate the need fo pope shielding by calculating the voltage that would appea on coil duing the fiing of coil if no shielding wee pesent. We have modeled a conical fast theta coil with inne adius inceasing fom 6cm to 7cm, and length 30cm. We have assigned a ise time of 3µs and have distibuted cuents so that a maximum field of 4T develops on axis. We then calculated the flux at each of the 3 wie locations of coil. Taking the time deivative of these flux values, and summing gives the maximum induced voltage on this coil, which we found to be appoximately.mv. By placing the shield between the two coils, and completing the same calculation, we find the induced voltage to be educed to 60KV. The plots shown ae magnetic flux contous with (top) and without (bottom) the shield pesent

11

12 Time Evolution of Magnetic Fields in a Plasma Tanslation Geomety Now that we have shown esults petaining to the time evolution of the field fom the slow coils, and the high voltage induced fom the fast coil, we next bing the entie system togethe and examine the tempoal field vaiation. We use the same geomety given above. We have also used the coil cuents which wee given in the pevious sections. Now, we allow the slow coil to fie and magnetic flux to diffuse into the line egion. Then, at nea maximum line flux, we set the fast coil to fie. In the plots to follow, we show the evolution of these fields fo seveal time steps. In the fist thee plots we see the slow diffusion of magnetic field into the line egion. Plot fou shows the final flux contous immediately befoe activation of the fast coil. Plot five gives flux contous at half maximum fast coil cuent. Finally plot six gives the field line configuation at peak fast coil cuent.

13 t=0.0008s t=0.006s t=0.004s t=0.0033s t= s t= s

14 Time Evolution of On Axis Magnetic Field Finally, we analyze the magnitude of the magnetic field along the axis of symmety. We have plotted the magnetic field stength vesus z-location fo the times shown in the six contou plots above. On Axis Magnetic Field We see that when the fast coil fies, the shielding allows vey 4 little field to ente the 3.5 line egion. We also 3.5 see a dop in field stength in the egion.5 between the fast coil 0.5 and slow coils, howeve, 0 the field stength emains nealy 50% of Location on z-axis that in the mio egion. Magnetic Field (T) t=.0008 t=.006 t=.004 t=.003 t= t=

15 Code is Used fo Hadwae Design on FRX-L Expeiment Ou code is cuently being used by membes of the P-4 Plasma Physics goup at Los Alamos National Laboatoy as a design tool fo magnetic coils and shielding in the FRX-L expeiment. In ode to chaacteize and optimize thei FRC plasmas, the plasma must be held axially stationay; mio fields ae used to accomplish this. Below we show calculations of magnetic flux contous and magnetic pessues fo diffeent end-mio geometies. The fist figues show magnetic quantities fo the case with no end plates. Flux Contous Magnetic Pessue

16 Flux Contous Magnetic Pessue Next, thin end plates ae added to the theta coil (small plates shown in blue at z~±0cm), notice that the mio coil causes field enhancement at 4.5cm but does not each 3.5 cm. Finally, as still thicke end plates ae added, we see well defined mio egions in both the flux contous and the pessue cuves. Hee, the mio effect eaches small adii, with enhanced fields at inside of 3.5cm. Flux Contous Magnetic Pessue

17 R-θ Code: Field and Inductance Calculations Non-Symmetic Z-pinch Geometies.

18 Motivation fo Calculations in R-θ Geomety Ou second code models conductos in a z-pinch geomety. We ae inteested in a flux compession expeiment, whee magnetic flux is compessed by an imploding line. Ou code was developed to calculate the cuent division in an inductive divide which is used to divet a potion of the main bank cuent onto a conducting had coe inside the line. The coe cuent ecombines with the main cuent on the outside of the line, and thus still dives the implosion. Schematics ae shown below. z Conducting had coe 5cm ~.5 cm Line Shunt and feed cuents ecombine on line suface Retun Cuent Imploded Line Flux Chambe Metal Feed Cuent Insulato Seveal Shunt ods split cuent L L Atlas Bank Cuent Atlas

19 Ou inductive divide assembly will ceate two cuent paths. Fist, cuent may flow along a cental cylinde which feeds cuent to the had coe. This cylinde will be suounded by a set of conducting ods o shunt inductos, which cay cuent diectly to the outside of the line. All conductos will have dimensions geate than one skin depth, and thus we cannot assume them to cay cuent unifomly. We theefoe model both cylindes and ods with many constituent conductos to account fo non-unifom cuent distibution. An example geomety is shown whee a lage cylinde and eight shunt conductos cay cuent to a cylindical etun conducto. The main pupose of the code is to calculate inductances so that we may detemine the amount of cuent that will be diveted to the had coe. Inductive Divide Modeling

20 Inductance Appopiate Wie Spacing can be Detemined In a code whee solid conductos ae modeled in a filamentay manne, it is impotant to detemine which wie spacing gives the best esults. In ode to detemine this we model coaxial cylindes, whee the inductance Coax Inductance: Ode of Magnitude Changes in Wie Radius is given by μ 0 R oute.44e-07 Lcoax = ln π R inne Ode of Magnitude.4E-07.40E-07.38E-07.36E-07.34E-07.3E-07 Inductance (50) Inductance (500) Inductance (000) Analytic Inductance (000) The cente od of the coax was modeled with a single post (since the cuent distibution is unifom in the geomety). The oute conducto is composed of eithe 50, 500, 000, o 000 wie elements. Fo each case, the adius of the filaments wee vaied by factos of ten. Zeo on the ode of magnitude axis coesponds to the adius whee the wies ae as tightly packed as possible without ovelap.

21 Plot Allows Detemination of Appopiate Wie Spacing All solutions ae linea on the logaithmic scale The lage the numbe of wies, the lowe the sensitivity to changes in wie adius All cuves intesect the analytic solution (ed line) with the same ode of magnitude change in adius. The adius to be used is found to be.5 = 0 = o 0 0 o Whee 0 is defined to be the coss sectional adius whee the wies touch but do not ovelap.

22 Small Pulse and Pototypical Inductive Cuent Divide Constucted CVR measues total flux Pot into flux chambe fo B-Dot pobe Line with flux chambe Shunt ods Retun Posts Feed 0 kv, 0 ka, τ /4 ~ 700 ns Numbe of Shunt posts can be alteed fom 0 to. Cuent division is calculated with B-dot pobes and cuent viewing esisto (CVR)

23 Ratio of Chambe Cuent to Total Cuent 0.9 Cavity cuent / Total cuent Numbe of Rods Pulse Code

24 Conclusions We have demonstated that one can faily accuately model the electomagnetic popeties of solid conductos with tightly packed wie aays. The capability to calculate eddy cuent induction, magnetic field diffusion, and the esultant field maps fo systems in both theta-pinch and z-pinch geometies has been demonstated. Acknowledgements We would like to acknowledge seveal helpful convesations with Jim Degnan, Mike Fese, and Glen Wuden. This wok was sponsoed at the Univesity of Nevada, Reno by DOE OFES Gant DE-FG0-04ER5475.