of the extracting grid will fall through its accelerating potential, while electrons(even those with the full energy of the cathode-anode voltage)are kept inside. The potential far downstream is essentially that of the neutralizer, if its electron-emission grid, in order to prevent backflow of electrons from the neutralizer through the or capacity is adequate this potential is seen to be set above that of the accelera accelerating system. In addition, by making the total voltage", VI, larger than the Net voltage",VN, the ion extraction capacity of the system is increased with no on lerated a third grid ( decelerator grid") is added to more closely define and control VN, and the neutralizer is set at approximately the same potential as this third grid It is difficult to analyze the three-dimensional potential and flow structures just described. It is however, easy and instructive to idealize the multiplicity of beamlets s a single effective one-dimensional beam the result is the classical Child-Langmuir space charge limited current equation The elements of the derivation are outlined below a)Poissons equation in the gap: dφ b) Ion continuity en v,=]= constant (2) c) Electrostatic ion free-fall: 2e(- Combining these equations, we obtain a 2n order, nonlinear differential equation for 9(x). The boundary conditions are 中(0)=0,中(X=d)= (4) In addition, we also impose that the field must be zero at screen grid This is because(provided the ion source produces ions at a sufficient rate) negative screen field would extract more ions which would increase the"in transit positive space charge in the gap. This would then reduce the assumed negative screen field and the process would stop only when this field is driven to near zero (positive fields would choke off the ion flux). At this point the grids are automaticall extracting the highest current density possible and are said to be space charge limited 16.522, Space P pessan Lecture 13-14 Prof. Manuel martinez Page 4 of 2516.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 4 of 25 of the extracting grid will fall through its accelerating potential, while electrons (even those with the full energy of the cathode-anode voltage) are kept inside. The potential far downstream is essentially that of the neutralizer, if its electron-emission capacity is adequate. This potential is seen to be set above that of the accelerator grid, in order to prevent backflow of electrons from the neutralizer through the accelerating system. In addition, by making the “total voltage”, VT, larger than the “Net voltage”, VN, the ion extraction capacity of the system is increased with no change (if VN is fixed) on the final velocity of the accelerated ions. In some designs, a third grid (“decelerator grid”) is added to more closely define and control VN, and the neutralizer is set at approximately the same potential as this third grid. It is difficult to analyze the three-dimensional potential and flow structures just described. It is however, easy and instructive to idealize the multiplicity of beamlets as a single effective one-dimensional beam. The result is the classical Child-Langmuir space charge limited current equation. The elements of the derivation are outlined below: a) Poisson’s equation in the gap: 2 i 2 0 d en = - dx φ ε (1) b) Ion continuity en v j = constant i i = (2) c) Electrostatic ion free-fall: ( ) i i 2e - v = m φ (3) Combining these equations, we obtain a 2nd order, nonlinear differential equation for φ (x ). The boundary conditions are () ( ) 0 = 0, x = d = -Va φ φ (4) In addition, we also impose that the field must be zero at screen grid: x=0 d = 0 dx ⎛ ⎞ φ ⎜ ⎟ ⎝ ⎠ (5) This is because (provided the ion source produces ions at a sufficient rate), a negative screen field would extract more ions, which would increase the “in transit” positive space charge in the gap. This would then reduce the assumed negative screen field, and the process would stop only when this field is driven to near zero (positive fields would choke off the ion flux). At this point, the grids are automatically extracting the highest current density possible, and are said to be “space charge limited