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The steelmaking and its shoppers have benefited drastically from the various major technological advances of the final thirty years. As their clients turn into ever extra caliber awake, even though, steelmakers needs to proceed their efforts to reduce damaging impurities, reduce in addition to regulate damaging nonmetallic inclusions and attain the optimal casting temperature, content material of alloying components, and homogeneity.
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Extra resources for Principles of Klystrons, Traveling Wave Tubes, Magnetrons, Cross-Field Ampliers, and Gyrotrons
The radial acceleration of the electrons is d 2r = − η Er dt2 (3-27) where Er is the electric field produced by the electrons. Using the relations given in Chapter 2 for the electric field, (2-3), and for the charge density, (2-7), the electric field at the beam boundary is Electron Motion in Static Electric Fields I 2 π b ε ouo Er ( b ) = − 27 (3-28) so the equation of motion for electrons on the beam boundary as a function of time is d 2b ηI = 0 − 2 dt 2 π b ε ouo (3-29) d 2b ηI = 0 − d z2 2 π b ε ouo3 (3-30) or, as a function of z, is Next, let 1/ 2 ⎡ ηI ⎤ A = ⎢ 3⎥ ⎣⎢ π ε o uo ⎦⎥ = 174 P (3-31) where P= I V (3-32) 3/2 In Chapter 5, it will be shown that the relation between current and voltage in most thermionic diodes, including the electron guns on klystrons and TWTs is given by (3-32).
Then, as the electric and magnetic forces accelerate the electron in the +y direction, the trajectory takes on the shape of a teardrop. yo + uo ωc Ey y 3 uo 2 Bz Ey yo uo Bz u r = o ωc 0 0 2 uo 3 uo 2 Ey Ey = 0 = uo Bz = uo 3 u 2 o ωc uo ωc u 3 o ωc x uo 4 ωc Figure 4-2 Electron trajectories for various electric field levels. The magnetic flux density, Bz, and initial electron velocity, uo, are constant. As the electric field is increased, the electron trajectories become cusp shaped and then somewhat sinusoidal.
In Chapter 7, Busch’s theorem will be used with the universal beam spread analysis to define the behavior of magnetically focused electron beams. Forces resulting from axial electron motions, z , and a radial component, Br, of the magnetic flux density will be included so that the θ force equation is ( ) z − e zB r Fθ = m 2 r θ + r θ = e rB (4-11) which can be written as z − zB r) = η r ( rB d 2 r θ dt ( ) (4-12) Multiplying by dt, the following can be obtained: η 2π ( 2 π rdrBz − 2π rdzBr ) = d ( r 2θ ) (4-13) To attach physical significance to this equation, consider a ring of electrons moving axially and expanding radially as shown in Figure 4-3.