2ev and hence the efficiency is m √Gf(o)do 2m VI Notice that the beam current IB is related to mi by IB=mi. We can therefore re- m( o f() d there each of the factors is less than unity and can be assigned a separate meaning (11) is the utilization factor",i.e. it penalizes neutral gas flow (12) the backstreaming efficiency" penalizes electron backstreaming (13)o f(o)do=ne, the"nonuniformity factor"is less than unity because of the nonuniform ion velocity It is clear that, since f(o)do=1, we want to put most of f(o )where Vo is greatest, namely, we want to produce most of the ionization near the inlet. In that case f()=8(o-1), and ne =1. A somewhat pesimistic scenario would be f(9)=1 d mi proportional to dr. i.e., ionization rate proportional to field strength In that 「√@×1×do 16.522, Space P pessan Lecture 18 Prof. Manuel martinez Page 3 of 2016.522, Space Propulsion Lecture 18 Prof. Manuel Martinez-Sanchez Page 3 of 20 ( ) 1 a i 0 i 2eV F = m f d m ϕϕϕ ∫ i (8) and hence the efficiency is ( ) 2 2 1 a i 0 a a i 2eV m f d m = 2mV I ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ϕϕϕ ⎝ ⎠ ⎝ ⎠ η ∫ i i (9) Notice that the beam current IB is related to mi i by B i i e I= m m i i . We can therefore re￾write (9) as ( ) 2 1 B a 0 mi I = f d I m ⎛ ⎞⎛ ⎞⎛ ⎞ η ϕϕϕ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ∫ i i (10) where each of the factors is less than unity and can be assigned a separate meaning: (11) u mi m ≡ η i i is the “utilization factor”, i.e., it penalizes neutral gas flow. (12) B a a I = I η , the “backstreaming efficiency” penalizes electron backstreaming. (13) ( ) 2 1 Ε 0 f d = ⎛ ⎞ ⎜ ⎟ ϕϕϕ η ⎝ ⎠ ∫ , the “nonuniformity factor” is less than unity because of the nonuniform ion velocity It is clear that, since ( ) 1 0 f d =1 ϕ ϕ ∫ , we want to put most of f (ϕ) where ϕ is greatest, namely, we want to produce most of the ionization near the inlet. In that case f = -1 () ( ) ϕ δϕ , and ηΕ = 1. A somewhat pesimistic scenario would be f =1 (ϕ) , namely dmi dx i proportional to dV - dx , i.e., ionization rate proportional to field strength. In that case 1 2 2 E 0 2 4 = ×1×d = = 3 9 ⎛ ⎞ ⎛ ⎞ η ϕϕ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫