lecture 3 -...

Lecture 3High Power Microwave Sources

EEC746

Tamer Abuelfadl

Electronics and Electrical Communications Department

Faculty of Engineering

Cairo University

Tamer Abuelfadl (EEC, Cairo University) Lecture 3 EEC746 1 / 21

Electron Dynamics

1 Overview of Uniform Field Focusing

Brillouin Flow

Scalloping

Conned (Immersed) Flow

2 Uniform-Field Focusing and Laminar Flow

The Beam Equation



Electron Dynamics


Outline


Brillouin Flow

Scalloping



The Beam Equation



Brillouin Flow

Brillouin Flow

Fr = m(r − r θ

2)

=−eEr − er θBz

On the beam envelope r = b,

Er =−I

2πε0u0b

b−bθ2 =

η I

2πε0u0b−ηbθBz

According to Busch's theorem, θ uη

2

(Bz −

r20r2Bz0

)As in the cathode region Bz0 = 0,θ =

η

2Bz = ωL ≡ Larmer frequency

b+ ω2Lb =

η I

2πε0u0b

For Brillouin ow the equilibrium beam radius (b = 0), a =1

Bz

√2I

πε0ηu0b.

Outline


Brillouin Flow

Scalloping



The Beam Equation



Scalloping


Brillouin Flow Problems

1 Flux density is too small.

2 Flux density is too large.

3 Beam is converging at

the entrance.

4 Beam is diverging at the

entrance.

5 Beam axis is oset from

the magnetic eld axis.

6 Beam axis is tilted with

respect to the magnetic

eld axis.


Brilliouin Flow Pros and Cons

Magnetic eld required is the lower for any other focusing system.

Beam is extremely sensitive to misalignment and perturbation.

Brilliouin ow can be very nearly achieved under laboratory conditions.

In practice, the magnetic focusing eld that is used, even when the

cathode is shielded, is greater than the Brillouin value. Reasons for

this include transverse electron velocities (from thermal and other

eects in the gun) and the increase in beam size that results from RF

modulation.

Outline


Brillouin Flow

Scalloping



The Beam Equation



The Beam Equation

r − r θ2 =−η

(Er +Br θ

),

where θ is given by Busch's Theorem,

θ =η

2

(B−Bc

r2cr2

)The electric eld on the beam boundary r = b,

Er (b) =−I

2πbε0u0,

The envelope equation is given by,

b−bη2

4

(B−Bc

b2cb2

)2

=−η

[−I

2πbε0u0+Bb

η

2

(B−Bc

b2cb2

)]b+bω

2L

(1− B2

c

B2

b4cb4

)− η I

2πbε0u0= 0


The Beam Equation

b+bω2L

(1− B2

c

B2

b4cb4

)− η I

2πbε0u0= 0

πb2cBc = πg2B =⇒ Bc

B

b2cb2

=g2

b2

Using the Brillouin radius a =1

ωL

(η I

2πε0u0

)1/2

,

b+ ω2L

[b

(1− g4

b4

)− a2

b

]= 0

Normalizing the radii, R =b

a, and Rg =

g

a,

R + ω2L

[R

(1−

R4g

R4

)− 1

R

]= 0


The Beam Equation

R + ω2L

[R

(1−

R4g

R4

)− 1

R

]= 0

Equilibrium radius Re corresponds to Re = 0, and is given by the following

equation,

Re

(1−

R4g

R4e

)− 1

Re= 0 =⇒ Re =

[1

2+

1

2

√1+4R4

g

]1/2


The Beam Equation

R + ω2L

[R

(1−

R4g

R4

)− 1

R

]= 0

So the solution near the equilibrium R = Re (1+ δ ) can be linearized as

follows,

δ +2Ωω2Lδ = 0, where Ω = 2− 1

R2e

δ = Acos(√

2ΩωLt)

+Asin(√

2ΩωLt)

R = Re

[1+Acos

(√2Ω

ωL

u0z

)+Asin

(√2Ω

ωL

u0z

)]


Outline


Brillouin Flow

Scalloping



The Beam Equation




Absence of ux in cathode region

−b0θ2B =−η

(Er +BBb0θB

),

θB =η

2BB

Presence of ux in cathode region

−b0θ2 =−η

(Er +Bb0θ

)θ =

η

2

(B−Bc

b2cb20

)b0θB

(ηBB − θB

)= b0θ

(ηB− θ

)B2 = B2

B +B2c

b2cb20


B2 = B2B +B2

c

b2cb20

Denition

The connement factor m is dened

as,

B = mBB

b2cBc = g2B = g2mBB , → g =

(1− 1

m2

)1/4

b0, → Rg =

(1− 1

m2

)1/4

Re

∵ Re

(1−

R4g

R4e

)− 1

Re= 0, ∴ Re = m,

Rg =[m2(m2−1

)]1/4

lecture 3 -...

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