RF Tutorial

G Burt

TM010 Monopole Mode

0 0

0 10

2.405

0

0

2.405

0

0

i tz

z

r

i t

r

rE E J e

R

H

H

i rH E J e

Z R

E

E

E

H

Beam

Z0=377 Ohms

Transit time factor• An electron travelling close to the speed

of light traverses through a cavity. During its transit it sees a time varying electric field. If we use the voltage as complex, the maximum possible energy gain is given by the magnitude,

• Where T is the transit time factor given by

• For a gap length, g.• For a given Voltage (=E0L) it is clear that

we get maximum energy gain for a small gap.

/ 2

/0

/ 2

,L

i z cz

L

E eV e E z t e dz E LT

/ 2

/

/ 2

/ 2

/ 2

, sin

,

Li z c

z

L

L

z

L

gE z t e dz

Tg

E z t dz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

g/bl

Tra

nsi

t ti

me

fact

or,

T

Transit time factor• The maximum energy gain is hence

• Considering the cavity length is g=l/4

0

sin g

E eV E Lg

0

sin 4E eV E

Shunt Impedance

• Another useful definition is the shunt impedance,

• This quantity is useful for equivalent circuits as it relates the voltage in the circuit (cavity) to the power dissipated in the resistor (cavity walls).

• Shunt Impedance is also important as it is related to the power induced in the mode by the beam (important for unwanted cavity modes)

21

2c

sc

VR

P

TM010 Shunt Impedance

0

0 10

sin 4

2.405

cV E

i rH E J

Z R

2

220

, 120

220

, 120

220

120

1

2

2.4052

2.405

2.405

c surface

c ends surface

c walls surface

c surface

P R H dS

E rP R r J dr

Z R

EP RL R J

Z

EP R R L R J

Z

2

231

sin 4

/ 4 2.405s

surface

RR R R J

Radius

• For a pillbox TM010 mode the radius is given by

• k=w/c• R=2.4/k=2.4l/2p

2

231

sin 4

2.405 2.405/ 4 2.405

2 2

s

surface

RR J

• What is the maximum gradient of a 100 cell 12 GHz structure driven by a 50 MW klystron (assume pillbox parameters)

• Rs=5E4/Rsurf

• For 100 cells

2Rsurf

4 46

8 7

5 10 2 5 10 21.25 10

2 (12 )(1.7 10 )(4 10 )sR

GHz

8(100 ) 100* 1.25 10s cells sR R

• What is the maximum gradient of a 100 cell 12 GHz structure driven by a 50 MW klystron (assume pillbox parameters)

• V for 50 MW

• Gradient is V/L

7 8 6(100 ) 5 10 1.25 10 78.8 10s cellsV PR

6679 10

63 100.0125 100

VGradient

L

= 63 MV/m