mathematical models and numerical investigation for the eigenmodes of the modern gyrotron resonators...

Mathematical Models and Numerical Investigation for the Eigenmodes

of the Modern Gyrotron Resonators

Oleksiy KONONENKO

RF Structure Development Meeting, CERN

2/36

Outline

Introduction

Mathematical model for the eigen TM modes

Mathematical model of the corrugated gyrotron resonator with dielectrics for the eigen TE modes

Mathematical model of the corrugated gyrotron resonator with dielectrics for the eigen TM modes

Discrete mathematical model of the hypersingular integral equation of the general kind

Numerical analysis of the gyrotron eigen modes

Conclusion

3/36

Coaxial gyrotrons as a part of thermonuclear facility

Introduction

4/36

Transverse and longitudinal cross-sections of the considered resonator

Eigen electromagnetic oscillations are considered Arbitrary corrugation parameters are studied

Introduction

5/36

Initial problem on the corrugation period

( )

2

2 2

, , , ,

( , ) ( , ) 0

( , ) 0

i z tz

S

E x y z t u x y e

u x y u x y

u x y

Propagation

constant

Cut-off wave number Frequency

2D Dirichlet problem:

ТМ modes

0zH

6/36

2D Helmholtz equation

1

1

0

21

22

2 2

2 2

, ,

, +2 ,

1 10

0

N

mm

imm m

m mm m

m S

m m

u r u r

u r e u r

u ur u

r r r r

u

Mode representation of the solution:

2D Helmholtz equation in polar coordinates:

Eigenvalue of the m-th TM mode

ТМ modes

7/36

Fourier-series expansion of the solution

21

2

, , ,

, , , sin

( ) ( ) ( ) ( ), , ,

( ) ( ) ( ) ( ) 2

n

i m nNTMm mn m nN m o m i m

n

TMm mn m i m i m n

n

TMn

u r A R R r e

u r B R h R r

J a Y c Y a J c na b c

J a Y b Y a J b

Expansions in the cross-cut domains:

Basis cylindrical functions expressions:

ТМ modes

8/36

Continuity condition on the domains boundary

21

,

, sin

( , , )1( , )

n

i m nNTMmn m nN m o m i

n

TMmn m i m i n

n

TMTM

c b

A W R R e

B W R h R

a b cW a b

b c

Electromagnetic field continuity means:

W functions can be expressed in the terms of the Φ ones:

ТМ modes

9/36

Hypersingular integral equation of the problem

2 2

2 2

2 2

2 2

2 2

12

2 3

: , , ( , )

1

( )

ln | | , , 0

m m i

m mm

m m m m m

F u R

F Fd d

F d K F d

The following unknown function is introduced:

Problem is reduced to the hypersingular integral equation (HSIE):

ТМ modes

10/36

Initial problem on the corrugation period

2D Neumann problem:

2

, , , ,

( , ) ( , ) 0

( , ) 0

i tz

S

H x y z t u x y e

u x y u x y

ux y

n

ТЕ modes/dielectrics

Eigen frequency

0 , 0zE

11/36

2D Helmholtz equation

1

1

0

21

22

2 2

, ,

, +2 ,

1 10

0

N

mm

imm m

m mm m

mS

m m

u r u r

u r e u r

u ur u

r r r r

u

n

Mode representation of the solution:

2D Helmholtz equation in polar coordinates:

ТЕ modes/dielectrics

Eigen frequency of the m-th TE mode

12/36

Fourier-series expansion of the solution

21

2

, , ,

, , , cos

( ) ( ) ( ) ( )( , , ) ,

( ) ( ) ( ) ( ) 2

n

i m nNTEm mn m nN m o m i m

n

TEm mn m i m i m n

n

TEn

u r A R R r e

u r B R h R r

J a Y c Y a J c na b c

J a Y b Y a J b

Solution expansions in the cross-cut domains:

Basis cylindrical functions expressions:

ТЕ modes/dielectrics

13/36

Continuity condition on the domains boundary

20

,

, cos

( , ) ( , , )

n

n

ikTEmn m nN m o m i

n

TEmn m i m i n

n

TE TE

A W R R e

B W R h R

W a b a b b

Electromagnetic field continuity means:

W functions can be expressed in the terms of the Φ ones:

ТЕ modes/dielectrics

14/36

Singular integral equation of the problem

2 2

2 2

2

2

( ) : ( , )

( , , ) 0

, 0

mm i

mm m

m m

uF R

r

Fd K F d

L F d

The following unknown function is introduced:

The problem is reduced to the singular integral equation (SIE) with the additional condition:

ТЕ modes/dielectrics

15/36

Discrete mathematical model of the SIE

2 2

( )

1

( ) 0

mm

TEn m n

V tF t

t

A v

To fulfill an edge condition Fm function is considered in such a form:

Discretization of the SIE is performed using quadrature formulas of the interpolative type based on the Chebyshev polynomials of the 1-st kind:

ТЕ modes/dielectrics

16/36

Initial problem on the corrugation period

2

, , , ,

( , ) ( , ) 0

( , ) 0

i tz

S

E x y z t u x y e

u x y u x y

u x y

Eigen frequency

2D Dirichlet problem:

0 , 0zH

ТM modes/dielectrics

17/36

2D Helmholtz equation

1

1

0

21

22

2 2

, ,

, +2 ,

1 10

0

N

mm

imm m

m mm m

m S

m m

u r u r

u r e u r

u ur u

r r r r

u

Mode representation of the solution:

2D Helmholtz equation in polar coordinates:

Eigen frequency of the m-th TM mode

ТM modes/dielectrics

18/36

Continuity condition on the domains boundary

21

21

, , ,

, , , sin

,

, sin

n

n

i m nNTMm mn m nN m o m i m

n

TMm mn m i m i m n

n

i m nNTMmn m nN m o m i

n

TMmn m i m i n

n

u r A R R r e

u r B R h R r

A W R R e

B W R h R

Solution expansions in the cross-cut domains:

Electromagnetic field continuity means:

ТM modes/dielectrics

19/36

Hypersingular integral equation of the problem

2 2

2 2

2 2

2 2

2 2

12

2 3

: , , ( , )

1

( )

ln | | , , 0

m m i

m mm

m m m m m

F u R

F Fd d

F d K F d

The following unknown function is introduced:

Problem is reduced to the hypersingular integral equation (HSIE):

ТM modes/dielectrics

20/36

1 12 2

2-1 -10 0

1 12 2

0 0 0

-1 -1

( ) 1- ( ) 1-1

( - ) ( - )

1ln | - | ( ) 1- ( , ) ( ) 1- ( )

u t t dt u t t dta

t t t t

bt t u t t dt K t t u t t dt f t

1 1

2 2 2 2

1 1

12

1

( , ) ( ) ( ) 1 ( ) 1 ( ) 1 1

( , ) ( ) ( ) 1

I

II

u v u t v t t dt u t t v t t t dt

u v u t v t t dt

Discrete mathematical model of HSIE

HSIE of the general kind

Inhomogeneous HSIE is considered:

In the polynomial spaces the following scalar products are considered:

21/36

12 2 2 2 2( )n n n n nA a bB K u f

1 2

0 20-1

12

0 0

-1

1 21

00-1

12

0 0

-1

1 ( ) 1-( )( )

( - )

1( )( ) ln | - | ( ) 1-

1 ( ) 1-( )( )

( - )

1( )( ) ( , ) ( ) 1-

u t t dtAu t

t t

Bu t t t u t t dt

u t t dtu t

t t

Ku t K t t u t t dt

Regularization of the integral operators

The following integral operators are defined:

The following regularized equation is considered:

Discrete mathematical model of HSIE

22/36

2

2

2

12 2

1 1 2

1

1

( )

( )

II

IIn

IIn

IIn

n

cB B

nc

nc K

K Knc f

f fn

2 In

cu u

n

Convergence of the discrete model

The following estimations of the convergence are derived :

Convergence of the approximate solution to the rigorous one :

Discrete mathematical model of HSIE

23/36

Discretization of the HSIE for the TM modes

22 ( ) 1

( ) 0

m m

TMn m n

F t V t t

A v

To fulfill an edge condition Fm function is considered in such a form:

Discretization of the HSIE is performed using quadrature formulas of the interpolative type based on the Chebyshev polynomials of the 2-nd kind for the regularized integral operators:

Discrete mathematical model of HSIE

24/36 Numerical investigation

Operating mode TE34,19

Frequency, f [GHz] 170

Number of the corrugations, N 75

Outer radius, Ro [mm] 29.55

Inner radius, Ri [mm] 7.86579

Depth of the corrugation, h [mm] 0.44

Width of the corrugation, L [mm] 0.35

Output power, P [MW] 2.2

Parameters of the ТЕ34,19 coaxial gyrotron

25/36 Numerical investigation

Gyrotron simulation software

26/36

Eigenvalue calculations for TE modes

0 10 20 30 40 50

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

TE

N

Relative accuracy of the TE34,19 mode eigenvalue calculations depending on the number of the discretization points

Numerical investigation

27/36

Eigenvalue calculations for the dielectrics and TE modes

1.0 1.5 2.0 2.5 3.0

100

110

120

130

140

150

160

170

180

190

Eigenvalue of thetraveling TE34,19

mode

TE 34,19

-

+

-

Dependence of the eigenvalue upon the dielectric permittivity

Numerical investigation

28/36

Field magnitude in the cross-cut

Real part of the Hz field component for TE34,19

mode

Numerical investigation

29/36

Absolute value of the Hz field component for TE34,19 mode in the corrugation

Numerical investigation

Field magnitude in the corrugation

30/36

Eigenvalue calculations for TM modes

Relative accuracy of the TM34,19 mode eigenvalue calculations depending on the number of the discretization points

0 10 20 30 40 50

1E-8

1E-7

1E-6

1E-5

1E-4 TM

N

Numerical investigation

31/36

Eigenvalues for a fixed azimuthal mode number

Eigenvalues of the ТЕ and ТМ modes for the fixed azimuthal mode number m=34

TM 34,18 TE 34,18 TM 34,19 TE 34,19 TM 34,20 TE 34,2099

100

101

102

103

104

105

106

107

108

109

110

TE TM

Numerical investigation

Mode SIE HFSS

TM34,18 100.1618 100.2053

TE34,18 101.8249 101.866

TM34,19 103.4942 103.5393

TE34,19 105.1494 105.1942

TM34,20 106.8159 106.8631

TE34,20 108.4674 108.5107

32/36

Eigenvalues for the cross-cut sets

Dependence of the eigenvalue upon the longitudinal z coordinate.

Problem is solved in each cross-cut separately.

0 10 20 30 40 50 60 70103.50

103.55

103.60

104.80

104.85

104.90

104.95 TM

35,19

34,19

z, mm

Numerical investigation

33/36

Field magnitude in the cross-cut

Absolute value of the Ez

field component for TM34,19 mode

Numerical investigation

34/36

Field magnitude in the corrugation

Absolute value of the Ez

field component for TM34,19 mode in the corrugation

Numerical investigation

35/36

0,0 0,5 1,0 1,5 2,00,00

0,02

0,04

0,06

0,08

0,10

0,12

L=0.31

, kW/cm2period

L=0.35

L=0.35

L=0.31

L=0.39

h, mm

Ohmic losses calculation

Estimation of the Ohmic losses denisity on the corrugation walls for the operating TE34,19 mode

ρ,kW/cm2 SIE IM

top 0.009 0

bottom 0.019 0.048

side 0.009 0.024

period 0.026 0.057

h=0.44

Numerical investigation

36/36

Conclusion

Mathematical model of the coaxial gyrotron resonator is developed for the eigen TM modes for the first time

Mathematical models to study gyrotron resonators with dielectrics are derived for TE and TM modes

Models are developed for the arbitrary corrugation parameters, radial and azimuthal mode indexes. This allows to use them for the analysis of the wide range modern gyrotron resonators.

New discrete mathematical model is built and substantiated for the hypersingular integral equation of the general kind. Numerical investigation of the TM waves was carried out on its basis. This model can also be used for other applied physics problems.

Basing on the developed models numerical analysis of the gyrotron resonators is performed. Comparison with the known results and validation is provided.

Results of the numerical estimation for the Ohmic losses density are presented and suggestions for the geometry optimization are proposed.

Conclusion

Thank you for your attention!