C
R
Lplasma
Lextra Lbank
Adjustable
(a) Coil (c) Representative circuit
(b) Mounting (d) Current traces
Figure D.6: (a) Simple construction of the coil. (b) Picture demonstrating how the coil is mounted on the plasma gun. The coils in the image are adjusted to be in the Large conguration. (c) Representation of the plasma discharge circuitry.
The system capacitance and resistance are C and R, respectively. The total system inductance can be changed by adding an adjustable Lextra. (d) The current prole for dierent coil congurations when the plasma bank is charge to 3kV.
the equation
VB = Apd ln(pd) +B
where V is the breakdown voltage, p is the pressure, and d is the gap distance. A and B are constants which vary depending on the gas used.
Cong Lext (nH) Ipeak (kA) tpeak (µs)
None 0 -33.9 4.8
Small 140 -30.2 6.1
Medium 210 -28.7 6.9
Large 290 -27.6 7.7
Max 370 -26 8.6
Table D.1: Measured value of the inductance
the coil connection points. Tabletop measurements using a LC meter show that the inductance scales linearly with the holes used to make the connection. Inductance measurement at four standard connection locations are summarized in Table D.1. The coils are mounted as shown in Fig. D.6 (b). The standard set-up uses two sets of coils placed in parallel with the plasma gun. Each set comprises two coils in series with the plasma gun so the equivalent inductance of the four coils is approximately that of a single coil. This approximation does not take into account coil-coil interactions and coil-chamber interactions but is expected to be accurate.
This additional inductance modies the current trace by increasing the charac- teristic frequency of the corresponding RLC circuit as shown in Fig. D.6 (c). The under-damped solution to an RLC circuit is given by
i(t) =Aexp(−αt) sin(t q
ω02+α2+δ) (D.1)
where α = R/2L is the damping factor and ω0 = 1/√
LC is the natural frequency.
Thus, an increase in L decreases the peak current and delays current peak by de- creasing the damping factor and increasing the natural frequency, respectively. This is shown in Fig. D.6 (d).
D.4.1 Current source
Adding Lextra ensures that the experiment acts like a current source. Kumar et al.
[104] performed eciency analysis on the Caltech Spheromak Experiment. They found that the combined impedances of the ignitron switch and the cables domi- nate the impedance of the plasma, and concluded that the spheromak experiment
Electrodes Electrodes
R a
a r
l
Single loop Spider legs
Figure D.7: Geometry of single plasma loop and Ref. [15] representation of eight spider legs.
circuit acts like a current source. While this assumption is valid for the spheromak experiment, it may not be a good assumption for the solar loop experiment.
The solar loop plasma inductance is likely larger than the spider-legs spheromak inductance. Kumar calculated the inductance by modeling the spheromak experiment current path as a co-axial cable. One way to compare the solar and spheromak inductance is to note that the spheromak experiment creates eight currying-carrying plasma loops in parallel (spider legs). This approximation estimates that the solar loop inductance may be up to eight times the spheromak inductance.
Consider the geometry shown in Fig. D.7. Suppose R= 4 cm anda = 1cm, the inductance of the loop (Eq. 2.2) gives ≈80nH. This can be estimated as 40 nH for a half circle or as 80 nH if image currents complete the circle. The inductance of the spider legs conguration [15] is
L= µ0l 2π lnr
a
(D.2)
which is like the inductance of a coaxial cable. Applying Eq. D.2 to spider legs with r = 4 cm, l = 4 cm, and a = 1 cm gives ≈ 10 nH of inductance, which is four to eight times smaller than the single loop inductance. Ref. [104] estimates the total series inductance in the discharge circuit (215 nH) is much greater than the plasma
(a) No Lextra (b) Maximum Lextra
Figure D.8: (a) Variation in the current prole for dierent strapping elds when Lextra = 0 nH. (b) when Lextra = 370 nH. Shaded region represents shot-to-shot variations.
inductance (30-50 nH) and concludes that the system acts like a current source. This assumption breaks down if single loop plasmas are four to eight times the inductance of the spider legs plasmas.
Kumar et al. [104] note that the ignitron inductance changes as a function of current. They report Lignitron ≈ 50 nH, 170 nH, and 800 nH at ∼ 80 kA, ∼ 10 kA, and ignitron turn-o, respectively. When calculating the total discharged circuit inductance (typically 215 nH), ignitron inductance (typically 170 nH) dominates other components inductance. Since the ignitron inductance is highly dependent on current, the system inductance may also be highly variable.
The addition of Lextra to the intrinsic system inductance (Lintrinsic in Fig. D.1) ensures that the total inductance of the discharge circuit is much greater than the single plasma loop inductance. This ensures that the plasma boundary conditions can be considered a current source. Figure D.8 shows variations in the current prole as a function of the strapping bank voltage. A higher strapping bank voltage means a stronger strapping eld, resulting in a smaller plasma with lower inductance. Lower plasma inductance correspond to higher peak current. The relative change in peak current for dierent strapping congurations is much larger whenLextra = 0 nH (Fig.
D.8 (a)) than when Lextra = 370 nH (Fig. D.8 (b)). The shaded regions around the
(a) Bipole (b) Coaxial
Figure D.9: Strapping eld lines (green) produced by (blue) coils (a) in bipole con- guration (b) in coaxial conguration.
current traces represent shot-to-shot variations for a given conguration.