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EFFECT OF THE ELIMINATION OF HIGHER-ORDER SPACE HARMONICS ON MAGNETRON CHARACTERISTICS
5.2.7 EFFECT O F THE ELIMINATION O F HIGHER-ORDER HARMONICS
O N MAGNETRON
SPACE
CHARACTERISTICS
by G . NOVICK and J. F. HULL
I. Introduction II. Anode Segment Face Design III. Experimental Results List, of Symbols References
I.
580 580 583 588 588
Introduction*
I t is t h e purpose of this section t o present a m e t h o d of design of t h e interaction space of a π-mode magnetron so as t o suppress t h e higher-order space harmonics a n d t o discuss some experimental work which was done t o t r y t o correlate t h e effects of these space harmonics with some of t h e oper ating characteristics of magnetrons. T o eliminate t h e higher-order space harmonics, t h e anode faces were properly shaped so t h a t t h e y could sustain only t h e fundamental forward a n d backward space harmonics. T h e experi m e n t performed w a s t o build t w o interdigital magnetrons, operating in t h e π-mode, as m u c h alike as possible except t h a t one magnetron h a d higher-order space harmonics in t h e interaction space a n d t h e other d i d not. On b o t h tubes t h e following operating characteristics were observed and compared: t h e variation of efficiency with operating voltage, t h e vari ation of efficiency with operating current, t h e mode shift current, t h e performance plot, a n d t h e frequency pushing characteristic a t various values of dc magnetic field.
II. Anode Segment Face Design
I n a parallel plane magnetron operating in t h e π-mode, as shown in Fig. 1, t h e ζ a n d χ components of t h e electric field in t h e interaction space a r e given b y J
d
n
- - -
\
ßng/2
/sinh7«a
*This work was originally done at the Army Signal Corps Laboratories, Fort Monmouth, New Jersey. 580
5 . 2 . 7 ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
Vo Ex = — — d
Ζ
2 =n _ o o
ßn,
5 8 1
2 ι w „ -1 1) wo / s i n ™' ßnQ/A yx z / ( — ~ ) r r mcosh _ L i ^ n -j e ßn 2 \ / W / sinh yna
— ( - l ) ^ 7n
(2)
where η is a n odd integer only. V0 is t h e peak rf voltage between t w o ad jacent anode segments
ßn =
and since normally λ 0 / 2 » d, t h e n ANODE
mr/d. SEGMENTS
h - g -1
Z
CATHODE
FIG. 1 . A parallel plane magnetron with square anode faces.
Combining t h e forward a n d b a c k w a r d traveling space harmonics into standing space harmonics, t h e expressions for t h e ζ a n d χ components of t h e electric field a r e
Ε . - 2 * ϊ
( - 1 ) 0 · - « *
d n=i E m
=
- 2
£
d
£ n
( ™ηΙ™\ ή
\ ( -
nirg/2d
l)(n-D/2 /
=i
\
~
^n(,/d)x ) sinh n(ir/d)a
2
nwg/2d
cosh n(r/d)x ) s m h η(τ/ά)α
, d
v
'
»
d
,
where η is now a positive odd integer. L e t t h e desired anode shape be given b y χ = x(z) (5) Since t h e anode m u s t be perpendicular t o t h e electric field a n d only t h e fundamental space harmonic component is desired, t h e n t h e slope of t h e anode m u s t be perpendicular t o t h e fundamental component of t h e space harmonics, namely, dx dz
1_
^ / r, , Kx/Ez\n=i
= t a n h -Ί χ t a n - ζ d d
(6)
Integration of t h e last equation results in t h e expression for t h e desired anode shape sinh -, χ cos 3 ζ = C a a
(7)
582
G. NOVICK AND J . F. HULL
where C is t h e constant of integration. Choosing t h e equipotential plane of t h e anode t o pass t h r o u g h t h e point (χ = a, ζ = 0, d, 2d, . . .) which is consistent with t h e same basic magnetron design considerations, t h e n C = isinh 2 a
(8)
where t h e ( + ) sign is for t h e positive anode segments a n d t h e ( —) sign is for t h e negative anode segments, as shown in Fig. 2. I t should be pointed ANODE
SEGMENTS
I—
—Η "
CATHODE
FIG. 2. A linear magnetron whose anode faces are properly shaped for only the fundamental space harmonic.
out t h a t t h e peak rf voltage applied between t w o adjacent anode seg m e n t s in Fig. 2 is V\ where 4 π
sin *g/2d irg/2d
and where V0 was t h e peak rf voltage between square anode segments of dimensions g a n d d of Fig. 1. I t should be noted t h a t t h e ratio of t h e two rf voltages, Vi/V0, varies from 4 / π = 1.27 for g/d = 0 t o a min 2 i m u m value of 8/π = 0.811 for g/d = 1. Applying a similar t y p e of analysis t o t h e circular magnetron, as shown ANODE
SEGMENTS
CATHODE
FIG. 3. Definition of dimensions for the circular magnetron.
5.2.7 ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
583
in Fig. 3 , t h e following relationship is obtained for t h e desired anode shape (1)*
ΚγΓ-(?Π
ΝΘ cos — = A
(9)
where A is now t h e constant of integration, Ν is t h e t o t a l n u m b e r of vanes, and r c is t h e cathode radius, a n d N/2y> (2π/λ0)τΆ.
ANODE
CURRENT
(amps)
FIG. 4 . Performance plot of E S M - 1 5 - 3 magnetron with the square anode faces. Number of segments is 1 6 ; anode diameter 0.375 in.; cathode diameter 0.225 in.; Q z , = 5 2 ; V0 = 279 volts; and B0 = 3 6 4 gauss.
III. Experimental Results
T h e p r o t o t y p e of t h e t u b e s used in this experiment is described in a previous paper (β). A cathode decoupling choke w a s used so t h a t t h e magnetron operated in t h e cavity, or π-mode. One of t h e t u b e s of t h i s * H. Gutton derived an equation for the anode shape which would supposedly elimi nate all modes except the 7r-mode in a magnetron; however, the resulting equation (9) merely eliminates all space harmonics of the 7r-mode, except the fundamental.
584
G. NOVICK AND J . F. HULL
experiment, t h e ESM-15-3, h a d square anode faces as in a conventional magnetron a n d therefore h a d m a n y higher-order space harmonics; while the other tube, t h e ESM-15-6, was alike in all respects except t h a t it h a d rounded anode faces in accordance with E q . (9). A comparison of t h e performance plots of t h e t w o tubes shown in Figs. 4 and 5 shows no outstanding difference in mode shift current or b o u n d a r y IOOOW
GAUSS
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3
"
V
,οοΛ ^
I500W
^
^
^
77 = 5 0 %
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^
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^
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•^ ^ ^ ^
^ • "
y 30%
j /
1 1 3 0 - ^ — — - — ^ ^ — — =
1
— 7 3 0 ^ - ^
5
— ^
MODE
SHIFT
CURRENT
0.5-1
1
1
1——-I
1
1
o-\
1
1
1
1
1
1
h
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0
ANOOE
CURRENT
(amps)
FIG. 5 . Performance plot of ESM-15-6 magnetron with the shaped anode faces. Number of segments, anode diameter, cathode diameter, Vo, and Bo are the same as the E S M - 1 5 - 3 ; QL - 110.
of lowest operating anode voltage. However, there are marked differences between t h e efficiency characteristics of t h e t w o tubes, a n d Figs. 6 a n d 7 compare their electronic efficiencies as functions of operating voltage a n d anode current, respectively. T h e shapes of t h e curves of electronic efficiency v s operating voltages of t h e tubes are typical. T h e electronic efficiency of t h e t u b e with square anode faces a t higher anode voltages is higher t h a n t h a t of t h e shaped anode faces a t t h e higher voltages. Also, Fig. 7 shows t h e same thing for
5.2.7
ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
585
t h e higher anode currents. T h u s there is a tendency for t h e t u b e with t h e square anode faces t o h a v e a higher electronic efficiency t h a n t h e other one. I t is r a t h e r difficult t o imagine t h a t t h e space harmonics increase t h e efficiency, a n d therefore one is forced t o seek another explanation for t h e results.
eo Η
o-\
ο
1
1
0.5
1.0
"—|
1
1
1.5
1
ANODE
VOLTAGE
1
1
2.0
1
1
2.5
h
Τ
3.0
(kv)
FIG. 6. Electronic efficiencies of the two magnetrons vs anode voltage.
One such explanation is t h a t because of t h e effectively wider gaps be tween segments in t h e shaped anode face t u b e , t h e electron trajectories p e n e t r a t e further radially in t h e open regions a n d t h e electrons lose t h e axially confining influence of t h e end h a t s . Therefore, more electrons escape axially in t h e shaped anode t u b e , especially a t t h e higher anode currents where t h e radial velocities of t h e electrons are greatest. I n addition, t h e local disturbances of t h e dc electric field b y t h e gaps
586
G. NOVICK AND J . F. H U L L
between segments m u s t be greater for t h e rounded segments, which in creases t h e double frequency force on t h e electrons due t o this effect a n d could reduce t h e electronic efficiency. Also, because of incidental difference in loading between t h e t w o tubes, t h e segment-to-segment impedance was different, and this could also affect t h e electronic efficiency, although this
eoJ
1 1 1 1 1
τ — ι — ι —
— ι — ι — Γ
1
γ-SQUARE S
S^
~
60
-
\ V
ANODE
FACES
( B = 1 4 0 0 gauss)
X
-
μ
1
1
£ uj 5 0 ο u_
\
.
Ü. 40
-
Ε ο or Ü UJ
SHAPED
ANODE
FACES'^
\ χ
( Β = 1 4 0 0 gauss)
30-1
1
1
j
1
\
|
UJ 20
10
οΗ ο
1
1
1
1
1
0.2
0.4
0.6
0.8
1.0
ANODE
CURRENT
τ 1.2
(amps)
FIG. 7. Electronic efficiencies of the two magnetrons vs anode current.
effect is typically small for C W magnetrons of this type. I t can b e con cluded t h a t space harmonics do n o t adversely affect t h e electronic effi ciency of magnetrons. One could say also t h a t space harmonics do n o t strongly affect t h e mode shift current, since t h e increase of t h e mode shift current in t h e rounded segment face t u b e was about w h a t would be ex pected because of t h e higher loaded Q. T h e greatest difference between t h e tubes, however, is in their pushing
5.2.7 ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
1990 GAUSS
ν
y
Y J-
261 5 1825 GAUSS
ε
ζ
587
ν y
'
y
1650 GAUSS
2605
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^
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Υ
^
^
^
~
0
'
^
0
^
^
~~~~
S **- 2 6 0 0 - ^ ^ - ^ ^ - 1 4 0 0
~
1130
GAUSS
GAUSS
2595-
-V ^
0.2
0.4
0.6
850
GAUSS
0.8
ANODE
1.0
1.2
1.4
CURRENT
(amps)
FIG. 8. Frequency pushing of the ESM-15-3 magnetron with square anode faces.
characteristics, as shown in Figs. 8 a n d 9. T h e changes in t h e frequency pushing characteristics of t h e t u b e s a r e considerably greater t h a n one would expect, d u e t o t h e incidental variation between tubes. While it is
2505 '
1
>
£
^
r-1825 GAUSS
\
2 5 0 0 — V
2
54
V
9
^
^
0.2
^
^
^
^—1130 1 0.4 ANODE
y 1
GAUSS
y/jfi
Syy
1
^ K ^ / ^
^ 2490 Η
r-1400
I
—
^ 6
05
G
AS U
S
GAUSS
Η0.6 CURRENT
1 0.8
1.0
(amps)
FIG. 9. Frequency pushing of the ESM-15-6 magnetron with shaped anode faces.
588
G. NOVICK AND J . F.
HULL
t r u e t h a t t h e frequency pushing of magnetrons is generally reduced as t h e loaded Q is increased, one would not expect as m u c h change in t h e pushing characteristics due to this effect as was observed. T h e p r i m a r y cause of frequency pushing is t h e variation of t h e effective phase angle of t h e b u n c h relative t o t h e space harmonic; and because of t h e proximity of t h e outer t i p of the bunch t o t h e circuit, its effect is relatively greater t h a n t h a t of t h e rest of t h e bunch. B u t t h e magnitudes of t h e higher-order space harmonics, relative to t h e fundamental, increase rapidly as t h e electrons a p p r o a c h t h e anode, and therefore t h e y m a y affect t h e phase angle of t h e t i p of t h e bunch and t h e pushing. List of Symbols
A, C a Β
d Ex, Ez g k Ν η r r& re Vo Vi χ, ζ ßn 7n θ λο Ql
constants of integration cathode-anode spacing operating dc magnetic field center-to-center anode segment spacing components of rf electric field segment-to-segment gap 2w/\o total n u m b e r of segments in a circular magnetron integer which specifies t h e order of t h e space harmonic radial coordinate in cylindrical geometry anode radius of circular magnetron cathode radius of circular magnetron peak rf voltage between adjacent anode segments peak rf voltage in t h e lowest order space harmonic rectangular coordinates propagation n u m b e r for t h e n t h harmonic 2
Vßn* - (2ττ/λ 0 ) angular coordinate in cylindrical geometry free space wavelength a t operating frequency loaded Q; this q u a n t i t y is proportional t o t h e ratio of t h e energy stored in the m a g n e t r o n cavities to t h e sum of t h e power lost in the magnetron cavities a n d t h e power delivered to t h e o u t p u t circuit References
1. U.S. Patent No. 2,610,309, H . Gutton, Sept. 9, 1952. 2. J. F. HULL and W. W. RANDALS, High power interdigital magnetrons. Proc. 36, 1 3 5 7 - 1 3 6 3 ( 1 9 4 8 ) .
1.RE.
O N MAGNETRON
SPACE
CHARACTERISTICS
by G . NOVICK and J. F. HULL
I. Introduction II. Anode Segment Face Design III. Experimental Results List, of Symbols References
I.
580 580 583 588 588
Introduction*
I t is t h e purpose of this section t o present a m e t h o d of design of t h e interaction space of a π-mode magnetron so as t o suppress t h e higher-order space harmonics a n d t o discuss some experimental work which was done t o t r y t o correlate t h e effects of these space harmonics with some of t h e oper ating characteristics of magnetrons. T o eliminate t h e higher-order space harmonics, t h e anode faces were properly shaped so t h a t t h e y could sustain only t h e fundamental forward a n d backward space harmonics. T h e experi m e n t performed w a s t o build t w o interdigital magnetrons, operating in t h e π-mode, as m u c h alike as possible except t h a t one magnetron h a d higher-order space harmonics in t h e interaction space a n d t h e other d i d not. On b o t h tubes t h e following operating characteristics were observed and compared: t h e variation of efficiency with operating voltage, t h e vari ation of efficiency with operating current, t h e mode shift current, t h e performance plot, a n d t h e frequency pushing characteristic a t various values of dc magnetic field.
II. Anode Segment Face Design
I n a parallel plane magnetron operating in t h e π-mode, as shown in Fig. 1, t h e ζ a n d χ components of t h e electric field in t h e interaction space a r e given b y J
d
n
- - -
\
ßng/2
/sinh7«a
*This work was originally done at the Army Signal Corps Laboratories, Fort Monmouth, New Jersey. 580
5 . 2 . 7 ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
Vo Ex = — — d
Ζ
2 =n _ o o
ßn,
5 8 1
2 ι w „ -1 1) wo / s i n ™' ßnQ/A yx z / ( — ~ ) r r mcosh _ L i ^ n -j e ßn 2 \ / W / sinh yna
— ( - l ) ^ 7n
(2)
where η is a n odd integer only. V0 is t h e peak rf voltage between t w o ad jacent anode segments
ßn =
and since normally λ 0 / 2 » d, t h e n ANODE
mr/d. SEGMENTS
h - g -1
Z
CATHODE
FIG. 1 . A parallel plane magnetron with square anode faces.
Combining t h e forward a n d b a c k w a r d traveling space harmonics into standing space harmonics, t h e expressions for t h e ζ a n d χ components of t h e electric field a r e
Ε . - 2 * ϊ
( - 1 ) 0 · - « *
d n=i E m
=
- 2
£
d
£ n
( ™ηΙ™\ ή
\ ( -
nirg/2d
l)(n-D/2 /
=i
\
~
^n(,/d)x ) sinh n(ir/d)a
2
nwg/2d
cosh n(r/d)x ) s m h η(τ/ά)α
, d
v
'
»
d
,
where η is now a positive odd integer. L e t t h e desired anode shape be given b y χ = x(z) (5) Since t h e anode m u s t be perpendicular t o t h e electric field a n d only t h e fundamental space harmonic component is desired, t h e n t h e slope of t h e anode m u s t be perpendicular t o t h e fundamental component of t h e space harmonics, namely, dx dz
1_
^ / r, , Kx/Ez\n=i
= t a n h -Ί χ t a n - ζ d d
(6)
Integration of t h e last equation results in t h e expression for t h e desired anode shape sinh -, χ cos 3 ζ = C a a
(7)
582
G. NOVICK AND J . F. HULL
where C is t h e constant of integration. Choosing t h e equipotential plane of t h e anode t o pass t h r o u g h t h e point (χ = a, ζ = 0, d, 2d, . . .) which is consistent with t h e same basic magnetron design considerations, t h e n C = isinh 2 a
(8)
where t h e ( + ) sign is for t h e positive anode segments a n d t h e ( —) sign is for t h e negative anode segments, as shown in Fig. 2. I t should be pointed ANODE
SEGMENTS
I—
—Η "
CATHODE
FIG. 2. A linear magnetron whose anode faces are properly shaped for only the fundamental space harmonic.
out t h a t t h e peak rf voltage applied between t w o adjacent anode seg m e n t s in Fig. 2 is V\ where 4 π
sin *g/2d irg/2d
and where V0 was t h e peak rf voltage between square anode segments of dimensions g a n d d of Fig. 1. I t should be noted t h a t t h e ratio of t h e two rf voltages, Vi/V0, varies from 4 / π = 1.27 for g/d = 0 t o a min 2 i m u m value of 8/π = 0.811 for g/d = 1. Applying a similar t y p e of analysis t o t h e circular magnetron, as shown ANODE
SEGMENTS
CATHODE
FIG. 3. Definition of dimensions for the circular magnetron.
5.2.7 ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
583
in Fig. 3 , t h e following relationship is obtained for t h e desired anode shape (1)*
ΚγΓ-(?Π
ΝΘ cos — = A
(9)
where A is now t h e constant of integration, Ν is t h e t o t a l n u m b e r of vanes, and r c is t h e cathode radius, a n d N/2y> (2π/λ0)τΆ.
ANODE
CURRENT
(amps)
FIG. 4 . Performance plot of E S M - 1 5 - 3 magnetron with the square anode faces. Number of segments is 1 6 ; anode diameter 0.375 in.; cathode diameter 0.225 in.; Q z , = 5 2 ; V0 = 279 volts; and B0 = 3 6 4 gauss.
III. Experimental Results
T h e p r o t o t y p e of t h e t u b e s used in this experiment is described in a previous paper (β). A cathode decoupling choke w a s used so t h a t t h e magnetron operated in t h e cavity, or π-mode. One of t h e t u b e s of t h i s * H. Gutton derived an equation for the anode shape which would supposedly elimi nate all modes except the 7r-mode in a magnetron; however, the resulting equation (9) merely eliminates all space harmonics of the 7r-mode, except the fundamental.
584
G. NOVICK AND J . F. HULL
experiment, t h e ESM-15-3, h a d square anode faces as in a conventional magnetron a n d therefore h a d m a n y higher-order space harmonics; while the other tube, t h e ESM-15-6, was alike in all respects except t h a t it h a d rounded anode faces in accordance with E q . (9). A comparison of t h e performance plots of t h e t w o tubes shown in Figs. 4 and 5 shows no outstanding difference in mode shift current or b o u n d a r y IOOOW
GAUSS
V"
3
"
V
,οοΛ ^
I500W
^
^
^
77 = 5 0 %
- ^ ^ ^ J
1820
1
0 6
-
4
ο
• ' 1.5-
Ο
1.0
77^^40%
^ t ^ " —
20
^
\
^
^>
•^ ^ ^ ^
^ • "
y 30%
j /
1 1 3 0 - ^ — — - — ^ ^ — — =
1
— 7 3 0 ^ - ^
5
— ^
MODE
SHIFT
CURRENT
0.5-1
1
1
1——-I
1
1
o-\
1
1
1
1
1
1
h
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0
ANOOE
CURRENT
(amps)
FIG. 5 . Performance plot of ESM-15-6 magnetron with the shaped anode faces. Number of segments, anode diameter, cathode diameter, Vo, and Bo are the same as the E S M - 1 5 - 3 ; QL - 110.
of lowest operating anode voltage. However, there are marked differences between t h e efficiency characteristics of t h e t w o tubes, a n d Figs. 6 a n d 7 compare their electronic efficiencies as functions of operating voltage a n d anode current, respectively. T h e shapes of t h e curves of electronic efficiency v s operating voltages of t h e tubes are typical. T h e electronic efficiency of t h e t u b e with square anode faces a t higher anode voltages is higher t h a n t h a t of t h e shaped anode faces a t t h e higher voltages. Also, Fig. 7 shows t h e same thing for
5.2.7
ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
585
t h e higher anode currents. T h u s there is a tendency for t h e t u b e with t h e square anode faces t o h a v e a higher electronic efficiency t h a n t h e other one. I t is r a t h e r difficult t o imagine t h a t t h e space harmonics increase t h e efficiency, a n d therefore one is forced t o seek another explanation for t h e results.
eo Η
o-\
ο
1
1
0.5
1.0
"—|
1
1
1.5
1
ANODE
VOLTAGE
1
1
2.0
1
1
2.5
h
Τ
3.0
(kv)
FIG. 6. Electronic efficiencies of the two magnetrons vs anode voltage.
One such explanation is t h a t because of t h e effectively wider gaps be tween segments in t h e shaped anode face t u b e , t h e electron trajectories p e n e t r a t e further radially in t h e open regions a n d t h e electrons lose t h e axially confining influence of t h e end h a t s . Therefore, more electrons escape axially in t h e shaped anode t u b e , especially a t t h e higher anode currents where t h e radial velocities of t h e electrons are greatest. I n addition, t h e local disturbances of t h e dc electric field b y t h e gaps
586
G. NOVICK AND J . F. H U L L
between segments m u s t be greater for t h e rounded segments, which in creases t h e double frequency force on t h e electrons due t o this effect a n d could reduce t h e electronic efficiency. Also, because of incidental difference in loading between t h e t w o tubes, t h e segment-to-segment impedance was different, and this could also affect t h e electronic efficiency, although this
eoJ
1 1 1 1 1
τ — ι — ι —
— ι — ι — Γ
1
γ-SQUARE S
S^
~
60
-
\ V
ANODE
FACES
( B = 1 4 0 0 gauss)
X
-
μ
1
1
£ uj 5 0 ο u_
\
.
Ü. 40
-
Ε ο or Ü UJ
SHAPED
ANODE
FACES'^
\ χ
( Β = 1 4 0 0 gauss)
30-1
1
1
j
1
\
|
UJ 20
10
οΗ ο
1
1
1
1
1
0.2
0.4
0.6
0.8
1.0
ANODE
CURRENT
τ 1.2
(amps)
FIG. 7. Electronic efficiencies of the two magnetrons vs anode current.
effect is typically small for C W magnetrons of this type. I t can b e con cluded t h a t space harmonics do n o t adversely affect t h e electronic effi ciency of magnetrons. One could say also t h a t space harmonics do n o t strongly affect t h e mode shift current, since t h e increase of t h e mode shift current in t h e rounded segment face t u b e was about w h a t would be ex pected because of t h e higher loaded Q. T h e greatest difference between t h e tubes, however, is in their pushing
5.2.7 ELIMINATION OF HIGHER-ORDER SPACE HARMONICS
1990 GAUSS
ν
y
Y J-
261 5 1825 GAUSS
ε
ζ
587
ν y
'
y
1650 GAUSS
2605
^
^
>
Υ
^
^
^
~
0
'
^
0
^
^
~~~~
S **- 2 6 0 0 - ^ ^ - ^ ^ - 1 4 0 0
~
1130
GAUSS
GAUSS
2595-
-V ^
0.2
0.4
0.6
850
GAUSS
0.8
ANODE
1.0
1.2
1.4
CURRENT
(amps)
FIG. 8. Frequency pushing of the ESM-15-3 magnetron with square anode faces.
characteristics, as shown in Figs. 8 a n d 9. T h e changes in t h e frequency pushing characteristics of t h e t u b e s a r e considerably greater t h a n one would expect, d u e t o t h e incidental variation between tubes. While it is
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FIG. 9. Frequency pushing of the ESM-15-6 magnetron with shaped anode faces.
588
G. NOVICK AND J . F.
HULL
t r u e t h a t t h e frequency pushing of magnetrons is generally reduced as t h e loaded Q is increased, one would not expect as m u c h change in t h e pushing characteristics due to this effect as was observed. T h e p r i m a r y cause of frequency pushing is t h e variation of t h e effective phase angle of t h e b u n c h relative t o t h e space harmonic; and because of t h e proximity of t h e outer t i p of the bunch t o t h e circuit, its effect is relatively greater t h a n t h a t of t h e rest of t h e bunch. B u t t h e magnitudes of t h e higher-order space harmonics, relative to t h e fundamental, increase rapidly as t h e electrons a p p r o a c h t h e anode, and therefore t h e y m a y affect t h e phase angle of t h e t i p of t h e bunch and t h e pushing. List of Symbols
A, C a Β
d Ex, Ez g k Ν η r r& re Vo Vi χ, ζ ßn 7n θ λο Ql
constants of integration cathode-anode spacing operating dc magnetic field center-to-center anode segment spacing components of rf electric field segment-to-segment gap 2w/\o total n u m b e r of segments in a circular magnetron integer which specifies t h e order of t h e space harmonic radial coordinate in cylindrical geometry anode radius of circular magnetron cathode radius of circular magnetron peak rf voltage between adjacent anode segments peak rf voltage in t h e lowest order space harmonic rectangular coordinates propagation n u m b e r for t h e n t h harmonic 2
Vßn* - (2ττ/λ 0 ) angular coordinate in cylindrical geometry free space wavelength a t operating frequency loaded Q; this q u a n t i t y is proportional t o t h e ratio of t h e energy stored in the m a g n e t r o n cavities to t h e sum of t h e power lost in the magnetron cavities a n d t h e power delivered to t h e o u t p u t circuit References
1. U.S. Patent No. 2,610,309, H . Gutton, Sept. 9, 1952. 2. J. F. HULL and W. W. RANDALS, High power interdigital magnetrons. Proc. 36, 1 3 5 7 - 1 3 6 3 ( 1 9 4 8 ) .
1.RE.