Applying Conformal Mapping to Derive Analytical Solutions of Space-Charge-Limited Current Density for Various Geometries

Harsha, N. R. Sree; Garner, Allen L.

doi:10.1109/ted.2020.3038619

Cited by 30 publications

(7 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Future experiments will be carried out to further validate the 3D injected beam simulation and the stability predictions. Additionally, a more complex, cylindrical crossed-field geometry will be studied theoretically using approaches applied to nonmagnetic field conditions [21,22], by simulation, and by experiment in future work.…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, these theories are often limited to planar 1-D geometries, not more realistic 3-D. Recent theoretical studies [21] applied variational calculus (VC) and conformal mapping to derive the first exact, closed form solutions to the 1-D space-charge limited current density in spherical [21] and cylindrical [21,22] coordinates; such an approach may ultimately provide a means to solve for instabilities in crossed-field devices with more realistic geometries. However, despite considerable progress, the perturbation problem in magnetrons [23,24] remains incompletely solved.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap

et al. 2021

Self Cite

View full text Add to dashboard Cite

This paper examines basic crossed-field device physics in a planar configuration, specifically electron beam perturbation and instability as a function of variation in magnetic field, and angle between magnetic and electric field. We perform a three-dimensional (3-D) simulation of electron perturbation in a planar crossed-field system using the full 3-D particle trajectory solver in CST Particle Studio (CST-PS). The structure has a length, height, width and anode-sole gap of 15 cm, 2 cm, 10 cm, and 2 cm, respectively. The anode to sole voltage is fixed at 3 kV, and the magnetic field and injected current varied from 0.01 T to 0.05 T and 1.5 mA to 1 A, respectively. The simulations show that applying a magnetic field of 0.05 T makes the beam stable for a critical current density of 94 mA/cm2 for an anode-sole gap of 20 mm. Above this current density, the beam was unstable, as predicted. Introducing a 1° tilt in the magnetic field destabilizes the beam at a current density of 23 mA/cm2, which is lower than the critical current density for no tilt, as predicted by our theory. The simulation results also agree well with prior one-dimensional (1-D) theory and simulations that predict stable bands of current density for a 5° tilt where the beam is stable at low current density (<13.3 mA/cm2), unstable above this threshold, and then stable again at higher current density, (>33 mA/cm2).

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…This expression is further used to evaluate SCL current, derived as (23) where the values of b, c, and d are calculated as 1, −1, and 1, respectively. On substituting ϕ = 0, voltage is calculated as…”

Section: Cylindrical Coordinate Systemmentioning

confidence: 99%

“…[10,22] Conformal mapping has been used to determine SCL current densities for 1D concentric cylinders, and for many other complicated geometries based on geometric arguments. [23] Lie-point symmetries have also been used to derive exact analytical equations for SCL current density for 1D spherical, tip-to-tip (t-t), tip-to-plate (t-p), and cylindrical diodes. [24] In recent works, [25][26][27] variational calculus (VC) techniques are suggested to calculate maximum current density using exact closed-form expression of CL law which is a big step further as compared to the numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

Coordinate System Invariant Formulation of Fractional‐Dimensional Child‐Langmuir Law for a Rough Cathode

Zubair

Fatima

Raheem

et al. 2023

Advanced Physics Research

View full text Add to dashboard Cite

This work presents an exact closed-form analytical model of the space charge limited (SCL) current density for planar, cylindrical, and spherical geometries in fractional dimensional spaces (F 𝜶 ) where 𝜶 represents the fractional dimensions having a range 0 < 𝜶 ⩽ 1. Decreasing value of 𝜶 shows an increasing degree of surface roughness. This analytical model approaches Langmuir-Blodgett equations for sufficiently close anode (R a ) and cathode (R c ) radii for integer dimensions (𝜶 = 1). To calculate SCL current density for any geometry, variational calculus (VC) is used to derive the differential equations in fractional (𝜶) dimensions. The effect of electrode radius ratios on space charge limited current is investigated here considering the surface roughness.

show abstract

“…In [5], a transit time model is applied to the cylindrical and spherical geometries and a new scaling laws was found. Harsha and Garner [7] applied conformal mapping to derive the analytical solutions of SCL current for various geometries. For high power applications and thermionic energy converters (TEC), outgassing and thermionic emission typically occur [34], [45].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Approach for Space Charge Limited Bipolar Flow in Cylindrical diodes

Liao

Yao

Zhu

2021

IEEE J. Electron Devices Soc.

View full text Add to dashboard Cite

In this paper, a numerical iteration approach to resolve the space charge limited (SCL) bipolar flow problem in cylindrical geometries has been developed. Such an approach is basis for the simultaneous determination of the unknown current densities and the potential distribution. We employed this method to study the characteristics of the SCL bipolar flow. By considering a cylindrical geometry with a cathode radius Rc and an anode radius Ra, the enhancement over the classical Langmuir-Blodgett (LB) law is investigated as a function of Rc/Ra. It is found that for the bipolar flow model, the SCL current density can be given by F×JLB, where F and JLB represent the enhancement factor on account of the influence of ions and the LB law, respectively. The enhancement factor F follows a Rc/Ra scaling and gradually converges to a constant with increasing Rc/Ra. The planar bipolar flow solution is recovered in the condition where the values of Rc and Ra are much greater than that of the gap spacing.

show abstract

Applying Conformal Mapping to Derive Analytical Solutions of Space-Charge-Limited Current Density for Various Geometries

Cited by 30 publications

References 20 publications

Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap

Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap

Coordinate System Invariant Formulation of Fractional‐Dimensional Child‐Langmuir Law for a Rough Cathode

A Numerical Approach for Space Charge Limited Bipolar Flow in Cylindrical diodes

Contact Info

Product

Resources

About