t.j. awe et al- magnetic field and inductance calculations in theta-pinch and z-pinch geometries

8/3/2019 T.J. Awe et al- Magnetic Field and Inductance Calculations in Theta-Pinch and Z-Pinch Geometries

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Magnetic Field and Inductance Calculat

in Theta-Pinch and Z-Pinch Geometri

T.J. Awe, R.E. Siemon, B.S. Bauer, S. Fuelling, V. MaUniversity of Nevada, Reno, NV 89557

S.C. Hsu, T.P. IntratorLos Alamos National Laboratory, Los Alamos, NM 87

NATIONAL LABEST. 1


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Physics Equations for RZ Geometry

Our first code, which we call EDDY, models cylindrically symmetric solid metawith tightly packed arrays of circular wire loops, each with circular cross section. only know three geometric parameters of the loop, namely the radius of the loopdistance of the loop center from the z=0 plane (z), and cross sectional radius of

(a). When calculating the vector potential, magnetic field, and mutual inductancean isolated loop, we assume the cross sectional radius (a) approaches zero, so tvalues are referenced to a single point, (r1,z1). When calculating the self inductresistance of the loop, the finite cross section must also be considered. We firs

vector potential at (r2, z2) due to a loop at (r1, z1),

cos2)(

cos

4),(

2

0 21

2

21

2

2

2

1

10

22 ++=

rrzzrr

dIrzrA

r

In order to dramatically reduce the computational time, we utilize fast elliptistatements. The vector potential becomes

( ) ( )221212

222

2

1022(

4

)(2)(2

2),(

zrrrrkwhere

kkEkKk

rr

IzrA

++=

=

r


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To calculate the mutual inductance between two loops we relate the field generatecurrent in loop one to the flux passing through loop two. We know

),(2 22212 zrArMI

r

==

where M is the mutual inductance between the two loops, thus we find

( )

= k

kEkKk

rrM

)(2)(2222

210

The resistance and self inductance of the loop above are found to be

2

2

a

b

A

LR ==

=

4

78ln

0 a

bbL

With this information, we may simply use the Kirchoff relation which states that shoconductors will have no change in potential as we traverse the loop. Thus we have

0=++ dtdIMIRdtdILj

j

ijiii

i

and

Here we have found n coupled differential equations that govern the transient ceach loop; this is precisely the type of problem that Matlab is suited to solve. Wein a position to find the current flowing through each wire element as a functionallowing for field calculations, including diffusion into the conductors.


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0.01 0.02 0.03 0.04 0.050.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

0 0.01 0.02 0.03 0.04 0.05-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Test 1: Driven Solenoid

Our initial test was to drive a 1KA solenoid

with 100 turns/meter. We expect to seeB=0.1257 T inside the solenoid, with the fielddropping to zero outside. The upper plotshows flux contours parallel to the axis anddecreasing in magnitude as we decrease inradius. This figure shows one problem withour simulation method. By using individual

wires to simulate solid objects, we havelocalized currents rather than a continuouscurrent density. Therefore, we find that nearthe loops the field is dominated by the local

current. This effect can be seen by thefringed contour along the surface of theconductor. In the second figure we look atthe magnitude of B as a function of radius.

Here the code has matched both qualitativeand quantitative expectation.


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0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

-800

-600

-400

-200

0

200

400

600

800

1000

Test 2: Coupling of Two W

Next we examine the most simplistieddy current induction. To the left wmagnetic flux contours for a perfecsource loop located at (.15, .02) and

inductively driven loop at (.1, .03). contours shown are for time t=5e-4 which is the specified rise time of the We see that the eddy current induc

shorted loop has the effect of limiting tflux through the plane of the loop. Toleft, we see the sinusoidal drive currealong with the induced eddy curreshorted loop shown in green. As expeddy current is of lower magnitude drive current, and there exists a slig

shift. The figure quantitatively agreesKirchoff relations previously stated.


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Test 3: Magnetic Field Diffusion

As the final and most rigorous test of code performance we use a driven solenoid to cr

Bz field, and analyze the diffusion of this field into a coaxial cylindrical conductor. The lospans radially from 2 to 2.8cm. We chose to analyze the axial component of the magn

two radial locations: A=1.75cm and B=3cm. At these points, we are far enough fromof the cylinder to be able to negate

the errors from the dominant localfields. We first test code convergenceby increasing the number of wirecolumns composing the shield from

1 to 2,3,4, and finally 8; convergencewas clearly shown. We then checkedthat the code was converging to thecorrect solution by comparing with

MHRDR, a trusted code. Thediffusion fields from the two codes areplotted to the right. When using at

least four columns the codes agreevery well, convincing us that EDDYcalculates field diffusion with

acceptable accuracy.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.00

Time (sec)

AxialM

agneticField(A

U)

Drive Field MHRDR 1 Colum n 2 Colum ns 4 Colum ns

Diffusion of Bz into Cylindrical Shield

IEddyIDrive

Bz(r,t)


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Next, we will calculate the eddy currents and fieldsin a geometry relevant to a plasma formation,translation, and implosion experiment. The

diagram shows generic geometric features of such

a system. We include the fast rising, high voltageconical theta coil for plasma formation and initialaxial acceleration, and the translation region

consisting of 6 slowly rising, low voltage, multi-turncoils. These coils produce a magnetic field whichslowly diffuses through the shield and liner,

creating magnetic flux inside the liner. Thespacing of the coils is reduced at each end,creating mirror fields which help to trap the plasma

in the liner region. We wish to examine severalfeatures of this geometry including the diffusion ofthe fields through the cylindrical liner and shield,the necessity of proper shielding to protect

experimental hardware, and the transient behaviorof the fields throughout the system.

X X

X X

x

x

x

Fast rising, hivoltage conic

theta coil fo

plasma formaand translatio

X X

X X

X X

X X

Aluminum Liner

M

Steel Shield

D

E

FC

B

A

Coil 1

Coil 2

Coil 3

Coil 4

Coil 5

Coil 6

Coil 7

.0508

0.10 m

.0

.05

.052

.06

.063

0.1016 m

.127 m

.127 m

.127 m

.033

Field Calculation of Formation/Translation

Region of Liner on Plasma Experiment


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Diffusion of Slow Magnetic

First, we analyze the diffusionmagnetic field from the slow coilsthe cylindrical conductors. Each ocoils consists of 32 evenly space

and each is driven as a perfect scurrent source with peak ampl20KA. The rise time of the coils is 2ms. At this frequency, we find thamore than half of the field manpenetrate through the conductors. see the effect of the mirror c

stronger fields exist at locations Dthan at location E. In the secondshow the time evolution of the magnetic field as a function of axis

We again see the increased field at the mirror locations.

On-Axis Magnetic Field Magnitude

0

0.5

1

1.5

2

2.5

3

3.5

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Location on z-axis (m )

MagneticField

(T)

t=0.6ms t=1.2ms t=1.8ms t=2.4ms t=3.0ms

Slow Field Diffusion Through Cylinders

0

1

2

3

4

5

6

7

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

Time (sec)

MagneticField(T)

F C E B D A


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Effects of Shielding

Next, we demonstrate the need for proper

shielding by calculating the voltage that wouldappear on coil 2 during the firing of coil 1 if noshielding were present. We have modeled a

conical fast theta coil with inner radius increasingfrom 6cm to 7cm, and length 30cm. We haveassigned a rise time of 3s and have distributedcurrents so that a maximum field of 4T develops

on axis. We then calculated the flux at each ofthe 32 wire locations of coil 2. Taking the timederivative of these flux values, and summinggives the maximum induced voltage on this coil,

which we found to be approximately 2.2MV. Byplacing the shield between the two coils, andcompleting the same calculation, we find theinduced voltage to be reduced to 160KV. The

plots shown are magnetic flux contours with (top)and without (bottom) the shield present.

0.05 0.1 0.15 0.6

0.7

0.8

0.9

1

1.1

0.05 0.1 0.15 0.6

0.7

0.8

0.9

1

1.1


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Time Evolution of Magnetic Fields in a Plas

Translation GeometryNow that we have shown results pertaining to the time evolutiofield from the slow coils, and the high voltage induced from the f

we next bring the entire system together and examine the tefield variation. We use the same geometry given above. We haused the coil currents which were given in the previous sections

we allow the slow coil to fire and magnetic flux to diffuse into tregion. Then, at near maximum liner flux, we set the fast coil tothe plots to follow, we show the evolution of these fields for seve

steps. In the first three plots we see the slow diffusion of magneinto the liner region. Plot four shows the final flux contours immbefore activation of the fast coil. Plot five gives flux contours

maximum fast coil current. Finally plot six gives the fieconfiguration at peak fast coil current.


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0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.110.6

0.8

1

1.2

1.4

1.6

1.8

0.01 0.02 0.03 0.04 0.05 0.06 0.00.6

0.8

1

1.2

1.4

1.6

1.8

t=0.0008s

t=0.00330t=0.0033012s

t=0.0024t=0.0016s

t=0.0033s


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Time Evolution of On Axis Magnetic Field

Finally, we analyze the magnitude of the magnetic field along the symmetry. We have plotted the magnetic field strength versus z-locathe times shown in thesix contour plots above.

We see that when thefast coil fires, theshielding allows very

little field to enter theliner region. We alsosee a drop in fieldstrength in the region

between the fast coiland slow coils, however,the field strength

remains nearly 50% ofthat in the mirror region.

On Axis Magnetic Field

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.6 0.8 1 1.2 1.4 1.6

Location on z-axis

M

ag

netic

Field

(T)

t=.0008 t=.0016 t=.0024 t=.0032 t=.0033024


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Code is Used for Hardware Design on FRX-L Experim

Our code is currently being used by members of the P-24 Plasma Physics groAlamos National Laboratory as a design tool for magnetic coils and shielding in experiment. In order to characterize and optimize their FRC plasmas, the plasmheld axially stationary; mirror fields are used to accomplish this. Below we show c

of magnetic flux contours and magnetic pressures for different end-mirror geomefirst figures show magnetic quantities for the case with no end plates.

Flux Contours Magnetic Pressure


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Next, thin eare added to

coil (smalshown in z~20cm), nthe mirror cfield enhanc

4.5cm but reach 3.5 cm

Flux Contours Magnetic Pressure

Flux Contours Magnetic PresFinally, as still thickerend plates are added,we see well definedmirror regions in both

the flux contours andthe pressure curves.Here, the mirror effectreaches small radii,

with enhanced fieldsat inside of 3.5cm.


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R- Code: Field and Inductance Calculations

Non-Symmetric Z-pinch Geometries.


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Motivation for Calculations in R- Geometry

Our second code models conductors in a z-pinch geometry. We are intereste

compression experiment, where magnetic flux is compressed by an imploding code was developed to calculate the current division in an inductive divider whito divert a portion of the main bank current onto a conducting hard core insideThe core current recombines with the main current on the outside of the liner, an

drives the implosion. Schematics are shown below.

z

Atlas Bank Current

Conductinghard core

Insulator

Metal

Liner5cm

L1 L2

~ 1.5 cm

z

Atlas Bank Current

Conductinghard core

Insulator

MetalMetal

Liner5cm

L1 L2

~ 1.5 cm

Atlas

ReturnCurrent

Several Shuntrods split

current

FeedCurrent

Shunt and feedcurrents

recombine onliner surface


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Our inductive divider assembly will createtwo current paths. First, current may flowalong a central cylinder which feeds currentto the hard core. This cylinder will besurrounded by a set of conducting rods or

shunt inductors, which carry currentdirectly to the outside of the liner. Allconductors will have dimensions greaterthan one skin depth, and thus we cannot

assume them to carry current uniformly.We therefore model both cylinders and rodswith many constituent conductors toaccount for non-uniform current distribution.

An example geometry is shown where alarge cylinder and eight shunt conductorscarry current to a cylindrical returnconductor. The main purpose of the code isto calculate inductances so that we maydetermine the amount of current that will bediverted to the hard core.

Inductive Divider Modeling


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Appropriate Wire Spacing can be Determined

1.32E-07

1.34E-07

1.36E-07

1.38E-07

1.40E-07

1.42E-07

1.44E-07

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Order of Magnitude

Inductance

Inductance (250) Inductance (500) Inductance (1000) Analytic Inductance (2000)

In a code where solid conductors are modeled in a filamentary manner, it is im

determine which wire spacing gives the best results. In order to determine this we mo

Coax Inductance: Order of Magnitude Changes in Wire Radius

The center rod of the

modeled with a s

(since the current disuniform in the geomouter conductor is co

either 250, 500, 100wire elements. For the radius of the filam

varied by factors of tethe order of magn

corresponds to the rathe wires are as tigh

as possible without ov

=

coaxLln

2

0

cylinders, where the iis given by


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Plot Allows Determination of Appropriate Wire Spa

All solutions are linear on the logarithmic scale

The larger the number of wires, the lower the sensitivity to

changes in wire radius

All curves intersect the analytic solution (red line) with the samorder of magnitude change in radius. The radius to be used isfound to be

10

105.0 o

o

rrr ==

Where r0 is defined to be the cross sectional radius where the

touch but do not overlap.


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Small Pulser and Prototypical Inductive Current Divider Constr

20 kV, 10 kA, 1/4 ~ 700 ns

Number of Shunt posts can befrom 0 to 12. Current divcalculated with B-dot probes andviewing resistor (CVR)

CVRmeasurestotal flux

Return Posts

Shunt rods

Liner with fluxchamber

PcB


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Ratio of Chamber Current to Total Current

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Number of Rods

Cavitycurrent

/Totalcurre

nt

Pulser Code


24/24

Acknowledgements

We would like to acknowledge several helpful conversations with JimDegnan, Mike Frese, and Glen Wurden.

This work was sponsored at the University of Nevada, Reno by DOE

OFES Grant DE-FG02-04ER54752.

Conclusions

We have demonstrated that one can fairly accurately modelectromagnetic properties of solid conductors with tightly packearrays. The capability to calculate eddy current induction, magnet

diffusion, and the resultant field maps for systems in both theta-pinz-pinch geometries has been demonstrated.

t.j. awe et al- magnetic field and inductance calculations in theta-pinch and z-pinch geometries

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