Current Measurements

A Rogowski coil is a device commonly used to measure AC current. It is usually constructed by helical windings of wires about a exible length of plastic tubing. The tubing is then bent into a hoop shape and the wire is returned through the center of the coil so that both terminals are now at the same end of the hoop (Fig. D.17).

i

(t)

B

b

R

C

a

i

(t)

V

Conductor

Passive Integrator

Figure D.17: Rogowski coil with a built in passive integrator. This particular con- guration is eective at measuring current at high frequencies where i2RVcap but

not so high so thati2R Ldi2/dt, where L is the inductance of the Rogowski coil.

follows Faraday's law

ξRogowski=− ∂Φ ∂t =−A ∂B ∂t =−AN µ0 2πb ∂i1(t) ∂t

whereA=πa2 is the pick-up area of the Rogowski coil andi1(t)is the current within

the central conductor.

Rogowski coils are considered air-core devices since the wire is not wound about a metal form. Thus, the coil is not prone to saturation and has a linear response over a wide range of currents. The Rogowski coil does not electrically perturb the circuit since the coil is not in electrical contact center conductor. While the current carrying conductor is in the center of the Rogowski coil in Fig. D.17, the voltage produced is independent of the conductor position within the coil for frequencies below 1MHz

[138]. For f >1 MHz, the Rogowski coil introduces a non-negligible inductance into

the relevant circuit equation.

nected to a passive integrator (Fig. D.17) is described by ξRogowski+L di2(t) dt +i2(t)R+ Rt i2(t0)dt0 C = 0 (D.6)

where R and C are the resistor and capacitor of the integrator, respectively [139].

If the inductive drop across the coil is small (L(di2(t)/dt) i2R) and the voltage

drop across the capacitor is also small (Rt

i2(t0)dt0/C i2R) then the current owing

through the capacitor (i2(t)) is given by

i2(t) =− ξRogowski R =−AN µ0 2πRb ∂i1(t) ∂t

so the voltage across the capacitor is then

Vcap(t) = Rt i2(t0)dt0 C = N RC a2µ0 2b i1(t)

Appendix E

Hall magnetic sensing

E.1 Introduction

The measurement of magnetic elds is critical to the study of plasma behavior. Unlike many astrophysical plasmas, laboratory plasmas are directly accessible to diagnostics. The most common measurement techniques use magnetic pickup (B-dot) probes and Hall sensors [140], where the choice between B-dot probes and Hall sensors boils down to the timescale of the target. Caltech spheromak experiments use B-dot probes to measure plasma dynamics on the microsecond time scale [16]. In contrast, Pegasus Toroidal Experiment [141] and TEXTOR [142] use arrays of hall sensors to measure magnetic elds with time scales of tens-of-milliseconds. Other experiments [143] com- bine hall sensors and B-dot probes for magnetic measurements at both short and long time scales.

Magnetic elds are spatially dependent vector quantities. A exible mounting mechanism, capable of volumetric measurements, is necessary to completely describe the magnetic eld. There has been signicant progress in volumetric measurements using B-dot probes. The Magnetic Reconnection Experiment (MRX) rotates a rake- shaped B-dot probe array to obtain volumetric measurements [130]. The Large Plasma Device (LAPD) uses motorized actuators [144] to precisely place magnetic probes at target locations. LAPD creates highly reproducible plasmas, and obtains volumetric measurements by adjusting probe placement over many repetitions. In contrast, there is signicantly less progress for volumetric measurement using Hall

I I + + + + + + - - - - B E

Hall effect on negative charge carriers.

VH

t

w l

Figure E.1: Hall eect when the magnetic eld is perpendicular to the sensor. sensors.

Harding et al. [145] report one of the earliest attempts to measure the 3-D mag- netic eld with Hall sensors. The authors place three small InSb Hall plates, mounted perpendicular to one another, to measure 3-D magnetic eld in a liquid Helium-lled space. This approach measures magnetic eld at a single location, and has limited spatial resolution due to the size of the Hall plates. Bongard et al. [141] and Jeong-hun et al. [143] use an array of Hall sensors to measure a single magnetic eld component along one axis. Duran et al. [142] mount nine sensors on three perpendicular planes, to obtain 3-D magnetic measurements along one axis.

These single axis measurements systems can characterize highly-symmetrical mag- netic eld congurations, but they are insucient for experimental set-ups that lack symmetry. Caltech solar-relevant plasma experiments have complicated asymmetric magnetic elds [82]. In particular, the iron-core bias coils behave non-linearly for high current ows, making the magnetic elds dicult to model. The bias elds produced by the coils must also diuse through copper electrodes at some locations. The diusion introduces a delay between the timing of the magnetic eld inside the vacuum chamber, and the timing of the current pulse used to create the magnetic eld. Magnetic eld lines diusing through the copper electrodes peak later than magnetic eld lines passing through regions without copper, creating magnetic eld proles with non-trivial spatial and temporal dependence.