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