Ion Wind (EAD) Thruster: Thrust, Corona and Collector Grid

Corona onset. Use relative air density $\delta=(p/101.325\,\mathrm{kPa})(293.15\,\mathrm{K}/T)$ and Peek’s surface field $E_c=3\times10^6 m_s\delta[1+0.0301/\sqrt{\delta r}]$ in V/m with $r$ in metres. The curvature constant has units $\sqrt{\mathrm m}$. Drew and Pister (2017), p. 4, Eq. (6), gives the wire–plane form at reference density. For a single wire this yields $V_0=E_c r\ln(2d/r)$.

Emitter interaction. The grounded-plane image-charge solution is derived here. Define $G_{ii}=\ln(2d/r)$ and $G_{ij}=\tfrac12\ln[1+(2d/s_{ij})^2]$ for wire spacing $s_{ij}=|i-j|s$. Solve $G a=\mathbf1$. The line charge is $2\pi\epsilon_0 V a_i$ and, neglecting the small variation of external field around a thin wire, the local surface field is $V a_i/r$. Thus $V_{0,i}=E_c r/a_i$. A measured common onset can replace this calculation.

Townsend current. Each wire carries $I_i=C_i V\max(V-V_{0,i},0)$. Masuyama’s MIT thesis (2012), printed p. 24, Eq. (1.20), gives Cooperman’s wire–plane estimate $C=2\pi\epsilon_0\mu L/[d^2\ln(f_{geo}/r)]$. Here the isolated-plane choice is $f_{geo}=2d$; the derived array extension replaces the reciprocal logarithm by $a_i$, giving $C_i=2\pi\epsilon_0\mu L a_i/d^2$. This extension is an estimate, not Gilmore and Barrett’s experimental correlation. Alternatively set $C_i=c_{measured}L$ using a coefficient fitted for the actual layout and air conditions. Drew and Pister, p. 4, Eq. (5), supports the Townsend relation and need for calibration.

Force and power. With stage ratio $q$, stage $k$ carries $I_k=I_1q^{k-1}$. Sum these stage currents; do not multiply the voltage by the stage count. Ideal axial ion force is $F=\sum I_k d/\mu$; retained assembly thrust is $T=\beta F$. Electrode power is $P_{HV}=V\sum I_k$ and supply power is $P_{in}=P_{HV}/\eta_{HV}$. This is the drift-current force relation used by Masuyama and Barrett (2013); $T/P$ is a dimensional performance ratio.

Packing and grid sizing. Frontal area is $A=Lns$. Stack depth is $D=(N-1)h+d+t+r$. Densities are $T/A$ and $T/(AD)$. The square-aperture unit-cell open fraction is $[a/(a+w)]^2$. Complete cell counts along panel dimensions are $\lfloor(L-w)/(a+w)\rfloor$ and $\lfloor(ns-w)/(a+w)\rfloor$. Their openings define finite panel transparency; leftover margins are solid metal. Collector metal volume is $N t(A-A_{open})$. The planar space-charge benchmark is $9\epsilon_0(V/d)^2/8$ per stage (Drew and Pister, p. 3); it does not cap the model.

Collector-width screen. Across the array, $W=ns$ and the outer-wire centre-to-side-edge margin is $m=s/2$. If either onset or current uses a plane-field estimate, require $W\ge4d$ and $m\ge d$; otherwise both onset and current must be measured for the complete layout. The geometric screen is derived here from the normal self-image field of an isolated line charge. Truncating the infinite-plane induced charge to a strip gives the retained fraction $f=[H(m/d)+H((W-m)/d)]/\pi$, with $H(z)=\arctan z+z/(1+z^2)$. At the worst permitted corner $m=d$, $W=4d$, this fraction is 0.9022. This diagnostic supports the chosen dimensions; it is not an error bound on a finite-conductor solution, nor a validation of the current estimate. The span direction retains the long-wire approximation.

Interstage domain. The nearest surface clearance from a previous grounded collector to the next emitter is $g_b=h-d-t-r$. Require $g_b\ge2d$ for this unshielded repeated-wire estimate. Nelson and Drew, §III.B and §IV.C (PDF pp. 3, 5), discuss the earlier wire-geometry two-gap heuristic and failures from unwanted interstage discharge. The nominal bulk-field screen checks both $V/(d-r)$ and $V/g_b$; a single stage has no backward path. These checks do not calculate sparkover.

Positive, steady DC corona in near-ambient air; straight round emitters; thin-wire electrostatics and a plane-equivalent collector. Ion drift dominates air convection, with constant supplied ion mobility and negligible recombination. The active span is long compared with the gap.

Every stage has its own emitter and collector, with all stages at the same applied potential difference. The current ratio represents stage interaction; overall retention represents the complete assembly’s force losses. This repeated-stage layout is distinct from the three-electrode, decoupled ionization/acceleration arrangement studied by Masuyama and Barrett.

The default mobility of 2 cm²/(V·s) follows Drew and Pister’s force convention (p. 9). Default retention 0.7, stage ratio 1 and converter efficiency 0.85 are design assumptions. They are not experimental fits. The calibrated Townsend coefficient is per total emitter length at the entered operating conditions, including array interaction. Transparency sizes the grid; it is not multiplied into thrust a second time.

Successive stages are aligned, unshielded wire arrays at the same positive emitter voltage, each followed by a grounded planar grid. Stage spacing is measured between wire-centre planes; collector thickness and emitter radius are subtracted when evaluating the preceding-collector clearance.

Use the geometry current mode for preliminary sizing and calibrate current, onset and complete-stack retention before relying on absolute thrust. The image-charge onset calculation is a Laplace-field thin-wire approximation; its array-current extension does not solve space-charge transport. After the first wire emits, space charge may also shift the remaining onset thresholds. Long-span, sparse-wire assumptions become less accurate near ends, closely packed arrays and a collector whose grid pitch exceeds its distance from the emitters.

Peek’s isolated round-wire onset is typically an approximate prediction. The reference checks compare it with Masuyama’s wire–cylinder measurements to 10% without fitting the constants. Density correction uses a 20 °C, 101.325 kPa reference. Supply mobility for the actual humidity, temperature, pressure and field; this tool does not infer mobility from those conditions. If the ionization sheath is a significant part of the gap, $Id/\mu$ overestimates the useful drift contribution.

Collector drag, ion interception, reverse corona, leakage, finite-span fields and downstream aerodynamic interaction are represented only by calibrated coefficients and retention, or remain outside the model. A stage current ratio of 1 assumes independent electrical operation and does not establish independent airflow. The planar space-charge result is a comparison scale, not a universal bound for a nonuniform finite array. Neither it nor the nominal bulk-air-field warning predicts sparkover.

Voltage sweeps contain 301 directly calculated points and may extend beyond the stable corona range of real hardware. They are not operating-voltage limits. At zero current, displayed thrust-per-watt results use the off-state convention zero; the curve omits this undefined ratio. The drawing is schematic, with enlarged wire sections and grid ribs. The stack envelope excludes frames, insulation, wiring, converter and battery.

High voltage and corona byproducts require electrical protection and ventilation. Insulation coordination, ozone exposure and discharge stability must be assessed separately; the calculator does not perform those assessments.

The finite-collector limits apply whenever either Peek onset or the Cooperman current estimate is selected. Measurements must cover the same complete array, grid, air conditions and operating-voltage range. Changing only one coefficient does not replace the other plane-field approximation. The truncated-plane image-field diagnostic describes an isolated wire's self-image contribution; it does not solve charge redistribution on a finite plate, and a 90% diagnostic does not imply 10% thrust accuracy.

The two-gap interstage threshold is a preliminary spacing screen inherited from wire-to-cylinder guidance, not experimental validation of this wire-to-grid stack. Bent, shielded, offset or independently biased stages are outside the entered geometry. Retention must still include any backward-current force and aerodynamic losses; spacing alone does not establish independent discharge. At rejected overlaps the backward-field denominator is bounded by the emitter radius to keep evaluation finite.