This material is based upon work supported by the U.S. Department of Energy Office of Science under Cooperative Agreement DE-SC0000661, the State of Michigan and Michigan

State University. Michigan State University designs and establishes FRIB as a DOE Office of Science National User Facility in support of the mission of the Office of Nuclear Physics.

Yoshishige Yamazaki, Kenji Saito, Steve LidiaNational Superconducting Cyclotron and Facility for Rare Isotope

Beams LaboratoryMichigan State University

PHY905 Section 4Ion Accelerator Technology

Lecture 9

Microwave Electronics for

Accelerators

RF Acceleration System and Its High Power RF Components(Two Cavities being Energized by One RF Power Source)

Y. Yamazaki, Lecture 9, Slide 3

Accelerating

RF Cavity

3 dB Directional

Coupler (Divider)

Matched dummy load

Circulator

Fundamental Mode

Power Coupler (FPC)

Tuning Plunger

Charged Particle

Beams

Matched dummy load

Klystron

Wave Guide

Low Level RF (LLRF)

Control to be by Lidia

Forwarding Electromagnetic Wave

Reflected Wave

Reflected when one of cavities tripped

Or non equivalent cavity case

Vector analysis to be used here

Y. Yamazaki, Lecture 9, Slide 4

Maxwell’s Equations Again

Y. Yamazaki, Lecture 9, Slide 5

=

Charge Conservation Law is not independent (implied in Maxwell’s equations)

Gauss’ Theorem

(M0)

(M1)

(M2)

(M3)

(M4)

Maxwell’s Equations Again

Y. Yamazaki, Lecture 9, Slide 6

=

Charge Conservation Law is not independent (implied in Maxwell’s equations)

Gauss’ Theorem

(M0)

(M1)

(M2)

(M3)

(M4)

Home Work 1:

Derive Eq. (M0) from

(M1)-(M4)

Gauss’ Law (generalization of Coulomb’s Law) and Its Differential Form

Y. Yamazaki, Lecture 9, Slide 7

Biot-Savart Law and Its Differential Form

Y. Yamazaki, Lecture 9, Slide 8

Derivative of Electric Field should be a kind of Current, that is, Displacement Current

Y. Yamazaki, Lecture 9, Slide 9

In order to make this

vanish this red field is

necessary, then J has to

form a loop.

Faraday’s Law of Induction

Y. Yamazaki, Lecture 9, Slide 10

Magnetic flux

Wave Equations and Plane Wave

Y. Yamazaki, Lecture 9, Slide 11

Home Work 2:

Derive Eqs. (W1) and

(W2) from (M1)-(M4)

without space current or

charge

(W1)

Plane Wave:

(W2)

(W3)

(W4)

is a solution of (W1), if

𝛽2= 𝜀𝜇𝜔2

(W3) implies that at , that is, the electric field maximum of

is moving with a velocity of which is referred to

as a phase velocity. The phase velocity in vacuum is a light velocity of

(W5)

This β is a wave number, being different from v/c.

Plane Wave: Example of Transverse Electric and Magnetic Wave (TEM)

No cutoff frequency

Y. Yamazaki, Lecture 9, Slide 12

Magnetic Field H

induced by

Induction

Displacement current

to induce H

t = 0

What shall happen, if conductor plates are placed? Current and charges induced, shielding fields

Y. Yamazaki, Lecture 9, Slide 13

If normal

magnetic field

H varies,

electric field E

would be

induced, giving

rise to infinite

current at the

conductor.

Varying magnetic field should be

parallel to conductor surface.

Conductor plates

Electric field on conductor surface induces charges, which shield the electric field

Y. Yamazaki, Lecture 9, Slide 14

Conductor Cavity

Surface

charge density

(Coulomb/m2) Induced

Field Electric field should be normal to

conductor surface, since parallel

field induces infinite current.

Y. Yamazaki, Lecture 9, Slide 15

Magnetic field along conductor surface induces currents, which shield the field

Surface

current density

Conductor platesConductor

-

Another TEM Wave Example: Coaxial Waveguide without cutoff frequency

Y. Yamazaki, Lecture 9, Slide 16

Conductor plates

Transverse Electric (TE) Wave Example:Rectangular Waveguide

Y. Yamazaki, Lecture 9, Slide 17

TE01 TE02

This kind of practice needed for waveguide modification exemplified by

Y. Yamazaki, Lecture 9, Slide 18

TE01

One can make a cut like this with

the minimum amount of influence

on the fields.

The cut can be used for electric

field monitoring.

Cutoff

Y. Yamazaki, Lecture 9, Slide 19

(W1)

Substitute (W3) in (W1), and then

(W3)

(R1)

(R3)

where

(R2)

Suppose

and then

(R4)

Define

and then

: cutoff wave number

: cutoff wavelength

(R7)

(R6)

(R5)

(R8)

λ0: wavelength

λ : guide wavelength

λc: cutoff wavelength

If β2 is negative, β becomes imaginary,

resulting in damping wave

Home Work

Y. Yamazaki, Lecture 9, Slide 20

TE01-mode rectangular waveguide and coaxial waveguide are two

commonly used waveguides.

The rectangular waveguide cannot be used for the low frequency, which is

lower than its cutoff frequency.

1. Calculate the waveguide width of b of the TE01 mode rectangular

waveguide for the cutoff frequency of 350 MHz.

Practically speaking, the TE01 mode rectangular waveguide is mostly used

for the high power RF with a frequency higher than 350 MHz, while the

coaxial waveguide is mostly used for the lower frequency.

2. Why are the coaxial wave guide not used for high frequency, high power

RF? Provide two reasons.

Coaxial Waveguide and Circular WaveguideLongitudinal Electric Field to be used for acceleration

Y. Yamazaki, Lecture 9, Slide 21

Coaxial wave guide (TEM)

Circular wave guide (TM)

Remove this inner conductor

Then, longitudinal electric field to be used for

particle acceleration appear.

Terminate both ends, and then a cavity is

formed, where standing waves appear.

Wave Equation with Space Current

Y. Yamazaki, Lecture 9, Slide 22

Home Work 3:

Derive this equation from

(M1)-(M4) with current

Skin Depth

Y. Yamazaki, Lecture 9, Slide 23