GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS J.e. Sprtt PLP 924 September 1984 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private circulatin nly and are nt t be further transmitted withut cnsent f the authrs and majr prfessr.
GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS J. C. Sprtt In PLP 849, Rse and Kerst discussed circuitry fr prducing a flat tp n the Levitated Octuple magnetic field wavefrm. This nte summarizes their results and treats several cases nt cnsidered by them fr a general capacitive discharge int an inductive lad. The first case cnsidered by Rse and Kerst as shwn belw has a series parasitic LC circuit whse purpse is t add a third-harmnic cmpnent t the current I I : CIRCUIT A Slutins are sught which are f the frm 1 1 0 ( sin wt + S sin 3 wt) where t = 0 is taken as the time the switch clses. The quantity S determines the amunt f third harmnic. As can easily be shwn, S 9 represents the cnditin fr which I( t) = 0 at the time at which! ( t) 0 and thus represents a reasnable criterin fr best flattening. Fr S > 9 the wavefrm has a 'single peak at wt = n/2, and fr S < 9 it has tw peaks n either side f a dip at wt = n/2. The results f Rse and Kerst fr S = 9 and C 2 initially discharged are summarized in Table I.
-2- Rse and Kerst als cnsidered the case belw in which the parasitic LC is placed in parallel with the lad inductance L 1 : c. L. CIRCUIT B This circuit requires less capacitance in the parasite and desn't require ne t flat C2 r Ll, but it des require a larger L2 and a higher vltage capacitr fr C2 as shwn in Table I. A prblem with bth circuit A and circuit B is that the parasite rbs energy frm Ll and thus lwers the peak field. Rse and Kerst prpse charging C2 with an initial vltage (0.75 V C1 (0) fr circuit A and 2.4 VC1 (0) fr circuit B) t achieve the same peak current in 1, as wuld have existed in the absence f the parasite. This wuld require a separate switch in series with C2, hwever. Furthermre, fr bth cases, the vltage rating fr the parasitic capacitr is different frm that f the main capacitr bank. An alternative slutin, nt discussed in PLP 849 is t place the series parasite f circuit B n the ther side f the switch and charge it t the same vltage as the main bank as shwn belw: CIRCUIT C
-3 - This circuit has the fllwing desirable features: 1) Only ne switch is required. 2) A separate charging circuit fr C 2 is nt required. 3) Bth capacitrs and L1 have a cmmn grund. 4) All capacitrs can have the same vltage rating. S) One can change frm the flattest field t the highest field by shrting ut L2 Circuit C can be implemented by taking a prtin f an existing bank and adding an apprpriate series inductr. The required values are listed in Table t. The effect f varying L2 with the ther values fixed is illustrated in figures 1 and 2 respectively. Nte that fr values away frm the ptimum, the symmetry f the wavefrm is destryed. Rse and Kerst pint ut that this may be useful fr cuntering resistive lsses in the circuit, emulating a pwer crwbar. One shuld be cautius, hwever, because deviatins frm the ideal value in either directin generally causes in excess f 100% vltage reversal f the parasitic capacitr in the absence f resistive lsses. A real circuit cntains resistive lsses in bth inductrs and can be represented as fllws: R e R, C, L a III L, CIRCUIT D C a
-4 - Rather than attempt an analytic slutin, the ptimum values f L2 and C2 were determined by numerical slutin f I( t) with R2/L2 R I /LI fr = Q = WL 1 /R 1 = 1 and Q = 2, using the qualitative behavir f figures 1 and 2 fr guidance ( see Table II). The ptimum values were then fit t functins f the frm 1 + aiq. The results f this prcedure are given in Table I. Figure 3 shws the wavefrms fr (1) n parasite, ( 2 ) parasite calculated neglecting lsses, and (3) ptimized parasite. Often it is desirable t clamp the capacitr vltage at zer when it attempts t reverse. In the absence f resistive lsses, such a crwbar wuld hld the current II cnstant frever, and n parasite wuld be required. With resistance, a crwbar may still be desirable t prlng the current and prevent vltage reversal n the capacitr bank. Such a circuit is shwn belw: c. L, CIRCUIT E The additin f the dide acrss C 1 has n effect n the preceeding calculatins except t reduce the vltage reversal n the parasitic capacitr ( see Table I). Vltage reversal f C2 can be eliminated entirely by prviding it with its wn crwbar dide, and there will be n effect n II since the vltage acrss C2 reverses after the vltage acrss Cl has been clamped t zer. The resulting wavefrms are shwn as dashed curves n figure 3.
- 5 - As a design example, we will calculate the parasite that wuld be required t flatten the tridal field wavefrm n Tkaple II. Fr a charging vltage f 3000 vlts, the present circuit gives a tridal field as shwn in figure 4, where t 0 is 6.0 msec after the pulse begins. Frm the tridal field, ne can calculate the winding current frm II = 2.5 x 10 6 BIN where B is the field in teslas and N = 96 is the number f turns. Over the range f interest (0 < t < 2 0 msec), the current is adequately mdeled by a crwbarred RLC with C 1 = 0.031 F, Ll = 1.3 mh and Rl = 80 mq. Frm circuit E f Table I, ne can calculate w = 91 sec- I, C 2 0.060 F, L2 = 700, R2 = 43 mq, and vltage reversal = 19%. The cst f ptimally wund cpper cils is estimated t be $ 2 8/msec/meter f length (installed). Thus fr a ne - meter - lng cil, the cst wuld be $500. In practice, we wuld prbably want an inductr with varius taps s that ne culd ptimize the flattening (at 8kG fr L2 700 l and 5 kv charge) r increase the field (t 10 kg fr L2 < 100 and 5 kv charge). Figure 5 shws the resulting wavefrms fr varius values f L2 (fr cnstant R2/L2).
TABLE I CIRCUIT PARAMETERS FOR FLATTEST WAVEFORM CIRCUIT L +L A B C 1/2 1/2 L +L ) ( _ 1 ) 1/2 1/2 ( ) ( ) - 3L 3 1 C 1 3L 1 C 1 w 3L ( 3L -'='-L'--':2_ 1 L2C 1 C 1 2 L L-71 D ( _ 1 E _ ) 112 C 2 /C 1 0.75 0.244898 1.333333 4 - + 0.934 (--) 3 wl 1 R l 1.08 R 1 1.08 4 - + 0.934 ( -) 3 wl 1 L 2 /L 1 244898 0.75 0.75 3 - - 0.284 (_) 4 wl 1 R 1 0.793 3 - - 0.284 (_) 4 wl 1 R 1 0.793 I '" I % REVERSAL 32.9914% 134.7151% 100% R 1 0.745" % 100-89 ( ) wl 1 R 0.302 100-91.5 (1- _ 1 ) % wl 1
-7- TABLE II ALGORITHM FOR OPTIMIZING WAVEFORM NO: RAISE L 2 OR C 2 DIP? 1st PEAK HIGHER? LOWER L 2 YES: 2 nd PEAK HIGHER? LOWER C 2
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z It. O 11. It:i Oz u- w Z w" Cl I N t- W I 11. «I It U Cl Z Z Il w w Cl 11. N t- 0 wn X D O N N, '<t [OJ ci z FIG. 2
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TOROIDAL FIELD (KG) --.---------.--------- I m () ()< - jo ""'" 0 - C - n ; \rt (t1 7' < r- r "'- AI IX) II - 3 (f) II () :t: 0 I\) II Q - 0 6" "1l 0 7'\ l' d' r fl\ VJ -; -------- TOKAPOLE II TOROIDAL FIELD CIRCUIT l-y1a I i\i P.{-;NI--':: C == 031 F I --- 50 MSEC FULL SCALE 1.3E-03 H R =.08 OHMS FIG. 5