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Microwave Resonators
CHAPTER
3 Microwave Resonators Robert A. Strangeway
I. Resonator Basics One-Port Reflection Resonator
G
C f
~
Figure I. Parallel resonant circuit.
1
~°°= CLC
= resonant radian frequency
tooC QU --
G
- u n l o a d e d quality factor (.JO o
Y = G 1 + jQu t°o
Handbook of Microwave Technology, Volume I
= circuit a d m i t t a n c e . CO
61
Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
Robert A. Strangeway
62
T-Line 1 L
G
C f ~
Go
Figure 2. Losslesstransmission line terminated in a resonant circuit.
If to = to o + Ato, then in the vicinity of resonance, Ato << too, the approximate admittance expression is Y = G(1 + jQu 2 Ato/too).
G O = transmission line characteristic conductance /3 =
60 G
= coupling factor
(/3 > 1, o v e r c o u p l e d ; / 3 = 1, critically c o u p l e d ; / 3 < 1, u n d e r c o u p l e d ) 2Ato (/3 - 1) - J Q u ~ too
F =
2Ato ( / 3 + 1) + J a u ~
= reflection coefficient
too
u n l o a d e d Q = Qu external Q = Qe
tooC G too C
60 ~o C
loadedQ=QL=
1
1
QL OH
ae
+
(Go+G)
1
au
=l+fl
au =/3 ae IFol = reflection coefficient m a g n i t u d e at to o
(1 -trot) /3 = (1 +
IFol)
(/3 < 1)
63
3. Microwave Resonators or
(1 +
IFul =
IUol)
r e f l e c t i o n c o e f f i c i e n t m a g n i t u d e for
IFul- ( (/~ - 1)2+ 1 (/~ + 1) ~ + 1 IFel = r e f l e c t i o n c o e f f i c i e n t
QL measurement
m a g n i t u d e for
Qe m e a s u r e m e n t
r e f l e c t i o n c o e f f i c i e n t m a g n i t u d e for
OL m e a s u r e m e n t
/ ( ~ __ 1) 2 + ~2 Irel-- (/3 + 1) 2 "q-~2 IFLI =
]~L]-- ¢
(]~°]22+ 1)
Two-Port Transmission Resonator [23, pp. 240-241]
] T-Line G°l
G
.
I
C
T-Line 2 G°2
"~
Figure 3. Two-port transmission resonatorcoupled to two transmission lines.
oJoC
Ou =
unloaded resonator Q =
ael =
external Q of input circuit =
Oe2 =
e x t e r n a l Q o f o u t p u t circuit -
QL, =
loaded Q without port 2
G
~o C
Q1 oJoC
02
64
Robert A. Strangeway
QL1
O9oC G+Go I
QL2--
loaded Q without port 1
QL2 O,~l,~ =
auQel Qu+Qel
090C
QuQe2
G+Go 2
Qu+Qe2
l o a d e d Q with b o t h p o r t s
oJoC
QLI,~
G + Gol -+-Go2 1
QL1,2
+
1
-F~
Qel
Qu
1
Qe2
/31 = p o r t 1 coupling factor =
Go 1 G
/3 2 = p o r t 2 coupling factor =
=
Qu Qel
Go2 Qu = G Qe2
QU
QLI,2
=
+ ~1 q" ~2
1
S 1 = s t a n d i n g - w a v e ratio of p o r t 1 at r e s o n a n c e with p o r t 2 m a t c h e d
131 3'1-"
1+132
$1 = 3'1 if 3'1 > 1; o t h e r w i s e S 1 = 1/3'1 S 2 = s t a n d i n g - w a v e ratio of p o r t 2 at r e s o n a n c e with p o r t 1 m a t c h e d ~2 Y2
1+/31
$2 -- 72 if 72 >
1;
otherwise S 2 =
1/72
[To[ 2 = p o w e r t r a n s m i s s i o n coefficient at oJo
ITol2 [12, pp. 200-201].
.---
471 (1 + 3'1) 2
,
~1 -- 3'1 ~1
.---
4131132 (1 + fll + f12) 2
3. Microwave Resonators
65
Skin Depth,
w h e r e / z is the permeability and or is the conductivity (SI units for all variables).
2. Two-Wire Line Resonators Resonance occurs when L = h o / 4 (quarter wave) or h o / 2 (half wave) and the line is properly terminated, usually with a short termination (an open termination tends to radiate). Radiation and termination reactances [21, pp. 286-288] will change fo. a = transmission line attenuation constant /3 = transmission line phase constant. The unloaded Q = Qu for a transmission line with low losses is Qu = (/3/ 2 a ) [7, pp. 77-79]. An approximate Qu for a two-wire line resonator is Qu = (Ixd2°~o/8Rwa) [15, p. 150], where /
R w = surface resistance = ~/ Y
T/'f~ or
or = conductivity /x = permeability Loaded Q must incorporate termination losses [21, pp. 288-293]. Terman ([29, p. 1049) determined the Q for nonradiating two-wire copper lines as Q = 0.08877rfbJ, where b = d in Figure 4 and J is read from the graph in Figure 5.
K
L
I\
/ Short or
d
Open
Figure 4. Basic two-wire line resonator.
66
Robert A. Strangeway 1.0
- ~ ~
0.9
%%
//
0.8 03 0.6 zO.5 0.4
0.3
-
0.2 0.1 0
2
I
5
I0
b/.~
20
50
I00
Figure 5. Terman's J factor (ignore other factors) [29, p. 1049] [© 1934AIEE (now IEEE)].
3. Basic Coaxial Resonators [23, pp. 2 2 5 - 238]
I
I
I
I
I
2zo
b Figure 6. Coaxial resonator [23, p. 227]. Reproducedwith permission of Artech House, Inc.
67
3. Microwave Resonators
TEM mode resonant frequencies are as follows: 2Z o = nAo/2 , where n = 1, 2, 3 , . . . and Ao is the TEM wavelength. (Note. Here, Z o is the half cavity length, not characteristic impendance.) Note that significant losses perturb fo. Q is the unloaded quality factor; when n = 1,
1 b
AO
l+-
2Zo
4-~,
[23, p. 226],
a
In -
a
where 6 is the skin depth (see Section 1). Be warned that coaxial line resonators can support non-TEM modes. An approximate formula for the higher mode cutoff frequency, fc, in coaxial transmission lines is [5, p. 259] ¢
fc
7r(a + b ) '
where c is the free-space velocity of light, ~ is the permeability of the medium, and e is the permittivity of the medium. Foreshortened Coaxial Line Resonators (Reentrant Coaxial Cavity Resonators) 2r °
.]'
d
½
~'--
2 rt
"7"-I
Figure 7. Foreshortened coaxial line resonator dimensions [I 5, p. 143]. [The figure was adapted from
Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.]
The resonant frequency is fo = (1/27rx/Lc), where /xh r1 L= In 2re ro 2
C=e°
rrr____& d _ 4ro In
(
2d e C h 2 + (r 1 - ro) 2
)]
[10, p. 3491
68
Robert A. Strangeway
and e is the base of natural logarithms. Resonant frequency design charts are also given in Moreno [23, pp. 230-238] and are accurate to within a few percent for gaps that are not too large. For these resonators, the approximate Q is r 1
ln-Q6
2h
,t °
A°
ro
rI 21nm +h ro
1 1 ] m+-ro
[23, p. 229].
r 1
Xi et al. [53] analyze a coaxial reentrant cavity with a dielectric in the gap.
4. R e c t a n g u l a r
t
Waveguide
X,-
Cavities
2
1A±)'. c÷)', c÷ Q
Q~
MODE
TE~mn
TEOmn
(p= + q2)(p2 + q2 + r~)s/2 abe • 4 ,c[p=r 2 + (i)2 + q2)2] + bc (q2r~ + (1=3+ q2)2] + obr ~ (pt + Cl~)
(q2 + r2) s/2
abc T"
q2c (b + 2a) + r~b(c + 2a)
TE¢On
obc (p2 + ,2) ~/= 2 " p2c(a + 2b) + r2a(c + 2b)
TM~..
.bc. (p2 + q=) (!~ + q2+ ,2),/2 4
TM¢~o
i ) l b ( a + c) +q2a (b + c)
obc
(p2 + q2) s/2
2 " p=b(a + 2c) + q2a(b + 2c)
p=
~, a
q.
m b
r--
ii
h e = Resonant Free-Space Wavelength 6 -- Skin Depth Figure 8. Resonant wavelength and Q [28, Vol. I, p. 180]. Reproduced with permission of Artech House, Inc. Note that Bazzy [2] also provides many of the figures in this and the next chapter.
69
3. Microwave Resonators
E:.
l-7~
131,311 I-6
-.4 ~TM 130, T M 3 1 0
~r~
T(O31.TE3OI
221
"rico3
-_
?.2
1.5~
6T[OI2.1T.I02 ~1-5
TM220 1.4-121.211
1'(O21 . T ( 2 0 1 1.3 "-"
m2.o 1.2 - -
D2.1 m "r[Oll. T[IOI
I.I
m2-2
"I "I'M 1 2 0 , T M 2 1 0 --
B
m2.3 i 1.0
el
B2.4
R
EXPLANATION 0-9
-_
L
---2.5
_ 0.8
--
0.7
~
0-6
0.5
B2.6
TMIIO
--2-7
-
o
T(
O
T [ & T M Modes.
Modes
only.
0
T M Mode~ only.
--2-.
--2e
Figure 9. Mode lattice for square waveguide resonators [3, p 834] [© 1947 IRE (now IEEE)]
70
Robe~ A. Strangeway
.--, ..
I
I
¢!" .... ~ I f.;'-q!
A,I~I / ! II I .~a~.-F~-4
I
.~4-
o
~
IXi ~.24; i !I III ~ , - - ,41,,- - ,~
r 4_
If ~ - 4 - - - - - ,
TMlll
It I/
v
TM~12 (see note)
IY
I TMl13 / (me note) /z
-X
/[ ,-.-_'-_-~
. r_-_2", [,r~.;__IvV 1/':..~_:) 1/' TM121
TM122 (see note)
TM123 (see note)
TM131 (see note)
TM132 (see note)
TMI~ (see note)
Figure10. TM mode field pattern sketches. Solid lines, E; dashed lines, H [I 5, p. 127]. [The figures are adapted from Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
3. Microwave Resonators
71
I
,.~~~:s~,t /
VtS"_----.-----~'V TEIm
TEIu
TEI~
TE lm (see note)
TErn
TEsu
(see note)
(see no~)
TErn (see note)
TE m (see note)
~
x
A, ~ - - - - ' : "IA
/ ~:2-",~/I .
-
-
II / ~.--.~.--Jlll
V ~'-"--"Y
TE~a
Figure II. TE mode field pattern sketches, Solid lines, E; dashed lines, H [15, p. 125]. [The figures are adapted from Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
72
Robert A.
Strangeway
% +0.01
--100
--
- 90
-
- 80
- 70 I.-
z i
•
U
-60
| v >-
I-
z. u U
o'
Og o.
~
5o ~ i
_>
EXAMPLE -0.03 - . -
%
%
+0.01
.100
_
..40
-
-30
i m
.90 0 80
-0.04
-
-0.01
•70
-0.02
60 50
"
-0.03
-
-0.04 -0.05
-0.05
-0.06
40
30 2O 10 0
NOMOGRAPH IS NORMAUZED RELATIVE TO 25"C AND 60% REL. HUMIDITY. -0.06
-20
-10
0
Figure 12. Humidity-cavity resonant frequency nomograph [28, Vol. I, p. 185]. Reproduced with permission of Artech House, Inc. [MIT RacI Lab Series, Techniqueof Microwave Measurement, Vol. I I 1944, McGraw-Hill; reproduced with permission of McGraw-Hill.]
73
3. Microwave Resonators
5. C i r c u l a r
Cylindrical
Waveguide
Cavities
WAVELENGTH AND Q FOR TEem. OR TMCmn MODES
-tt
~,.=
2
L
I
÷(:),
~,o Resonant Wavelength in Free Space Xc. mth root of J'¢(X) = O for the TE modes Xs.. mth root of J {[ X) = O for the TM modes ,,
,,
MODE
TEcmn
I - (XCm)=]J [X~m+ p'r2] 31'
Xcm
TMcmn
TM~mo
VX2¢m + par:
(n > o)
2r ( l + r )
Xcm (n
:r(2 + r)
5 = Skin Depth
r = D
L
-- o)
p = nf
2
Figure 13. Resonant wavelength and Q [28, Vol. I, p. 180]. Reproduced with permission of Artech House, Inc. Note that the humidity-cavity resonant frequency homograph appears in Figure 12.
74
Robert A. Strangeway
TH~ FIaST TWENTY-FIVE ZEROS OF J . ( x ) AND J . ' ( x ) No.
Value
Mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1.841 2.405 3.054 3.832 3.832 4.201 5.136 5.318 5.332 5.520 6.380 6.416 6.706 7.016 7.016 7.501 7.588 8.016 8.417 8.536 8.578 8. 654 8.771 9.283 9.647
TE~ TMol TE2x TMll TEol TEsx TM21 TE4x TEl2 TMo2 TM31 TE61 TE22 TM12 TEo2 TE61 TM~t TE82 TM~2 TExs TEn TMo~ TM51 TE~2 TE81
Figure 14. Bessel function zeros [23, p. 222]. Reproduced with permission of Artech House, Inc. Note that Saad [28, Vol. 2, pp. 192-195] has a more extensive listing.
75
3. Hicrowave Resonators
mO
I TMO2--~ r-
1'9 ~-TM022 _~TEI22 -~TF..412 -~-TM021
n
0.5
+TE3~3 -~T[,OI3, TMII3
TMQ~O
=I~TF-121 [411
+Te:2~3
I-0
--~-.TMOI3 -~)-TF..113
4-TE312 -~TF..OI2 ,TM II 2
T(12~ --1"7 T(44
--
"FTM212
1'8
TM211
T M 2 1 - - r"
-~TM023 ,,~ -~-TEI23LT[413 m -I)-TM213
TM210
+T.o,2 +Tu,2
1-6 +TE311
--
1-5 TF.OII, TMII
~
1.4
D
D
L
~
TE,21t
TE3I-- - - 1.3
+TMOI I TMIIO 1"2
1T.OI ,'I'M I1--~
m2.0 +TE III
I X PL AN ATION
~I'I /
[
I'O TE21 m -- O.9
TMOI"
--
0"8 TMOIO
-- 0 " 7 T£11m -- 0 " 6 --'O'5 - - 2-S
Figure 15. Mode lattice for cylindrical resonators [3, p. 832]. [© 1947 IRE (now IEEE)].
76
Robert A. Strangeway
TM 022 TM 021 3O
TE 122TM 212 28 TE 412, 26
24
ii~ii?i~ 22
20
~
16
u
14
z o
12
0
D
×
~..
e
10
i-~ii.i.!i. M O D E CHART FOR RIGHT CIRCULAR C Y U N D E R
0
I
2
3
4
S
6
7
8
9
(D/L)'
Figure 16. Mode chart for cylindrical resonators [28, Vol. I, p. 181]. Reproduced with permission of Artech House, Inc.
O4
_
t,')
..j
~.t'-
=7 ClOHI3 ~1 9 N I 7 d n 0 3
y
)
rn
w
rn
~I
{3
~ ;?J
~
w
w
m oJ
CD
~r
~< "-
w
-< ~ 3:
N
N
r
JC:
-J
c:o~ ,(
~
~,
9-
I
~l
~< •
~
"<~
"~
~
~1:::11~10 ~ V 7 N : ) ~ I D
~:
'<"
--
-
0
N
d66
~
C
d
_
5
-coo m -
6
~
d
O 0 ~ -
~
r"
o o o
_ ( ~ w w w SINVISNOD
0
m
~
5
(~
d
~"
:3
m "E
w
<
om Z~(M
I~I
~z~
N
toO_
~_5 o
_
m 0
¢)
._c
x ¢)
>--
~-J ~'-~
@m
~'
~
~
c
"~
of-
~g 0
~
~.
~
-
.gm
u
-
}j
E °
,,
I--
Z
IE
o >wz ~I0 w r--
.o
0--
I0
z
U
w
~
I--~
_c:
~.o
~Z
EL
o
~ +-.
~
L2~ .j
w
=
Z
_
~
Ow
~
~w
,,,,,..
uJ
L.
.,
<._
~_ _E
~.
p.~
~
~
o ~__
~:)
z
/J
TMulmode
TMoxI mode
E M . . . . .
TMo~
TMom
~
TMo~
m
I
I TMI~
TMm
TMm
( m note)
( m nora)
%
I
' 1 4&_.
g 5 ~J-.'~-',.' .,:)_;)
TM212 (seenote)
TM:m (seenote)
TM~m (seenote)
Figure 18. TM mode field pattern sketches [15, pp. 134-135]. [The figures are adapted from
Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
I I t
I
t
,,
i
i ~ \
,
! /
.
I1,
I A
~
/t'
/t'
It It
h i
Jl ',
"~-E
(a) TEosxmode
(b) TEsss mode
TEos~
TErns
TEoaz (see note)
TEss2
TErn (see note)
TE~s (see note)
m
TE212
TEmx
TEzsz
(see note)
(see note)
Figure 19. TE mode field pattern sketches [15, pp. 130-131]. [The figures are adapted from Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
79
80
Robert A. Strangeway
6. Spherical Cavities [4, p. 380]
E
H
Figure 20. Field patterns for the lowest order modes in spherical cavities. (a) TE i i0 mode. (b) TM i i0 mode. [From A. B. Bromwell and R. E. Beam, Theory and Applications of Microwaves, © 1947 McGraw-Hill with permission of McGraw-Hill.]
a
=
spherical cavity radius
wavelength
h r --
resonant
/~r =
1.40a for TEll m modes 2.29a for T M l l m modes.
~'r --
7. Dielectric Resonators
l
Z
Figure 21. Dielectric resonator dimensions and magnetic field for the TE0i 8 mode [19, p. 2]. Reproduced with permission of Artech House, Inc.
3. Microwave Resonators
81
The approximate resonant frequency, free space in the TE01 ~ mode is a ~34
fo-
fo, of a
[ aL- +3"45 1
dielectric resonator (DR) in
(GHz)
[19, p. 3],
where a = radius of the DR in millimeters L = length of the DR in millimeters er
relative permittivity of the DR.
=
This equation is accurate to approximately 2% under the conditions 0.5 < (a/L) < 2 and 30 < e r < 50. More accurate and mathematically intensive calculation procedures which account for the environment around the resonator, such as a microstrip substrate and metal shields, exist (see, for example, [11, 17, 19, 26]). An example of a mode chart, from Pospieszalski [26, p. 235], is shown in Figure 22. Later literature improves the accuracy, ease of calculations, or both and includes references to several approximate and rigorous techniques [44, 45, 49]. The determination of the length of a DR for a given diameter and frequency can be calculated to within a few percent using Kajfez's procedure ([18]; also shown in [19, pp. 148-149]) for the Itoh and Rudokas
(fo'D)2"erl
i
[GHz cm]2! ~0
4.103
"
L1 SI='E
TE01 1 mode ~
2.1o3
er I> 500 er =10
.~j~" . / ~ I"~
.....-- ..._--'-
......
S1= .
...~
j
. . f
. . . .
:
s~=o.4 . . . . .
1o103
o
0.5 1.0
i
o
i
o
i
i
i
i
o
i
i
i
i
i~
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure 22. A DR mode chart [© 1979 IEEE].
112
Robert A. Strangeway
I -
o
L
i ....
. JI
.
Er
E2 .
.
.
.
.
.
1
Ib ID Ill
T L2
t ......... mm
--
Figure 23. DR geometry and dimensions, e 2, is the substrate relative permittivity [18, p. 133] [© by Argus Business, reprinted with permission].
model [17]. Refer to Figure 23 for the geometry and variable definitions. Note the metallic cover at distance L 1 above the resonator. The straightforward calculations are listed below. Specify fo. Select D: 5.4
5.4
ko~r
< D < ~koCe2 ,
where
k o = free-space propagation constant = toV/~oe o Yo=
ko- ~
(e r -
1)-2.4052
2
Yo
h-~
.405+
(
2.43 ) 2.405 1 + ~ + 0.291y o
Yo fl
=
Ck oF, 2 r
--
h2
al= Ch2-k 2 Ot 2
L
=
=
Ch 2
-
_
2 2 koe
/3 t a n - 1 ~/3 coth al L 1 + tan - 1 ~a~2 coth a2L 2 -
.
83
3. Microwave Resonators
See [36, 39, 47] for dielectric ring resonators. See [37, 38, 40, 48, 51] for whispering gallery mode dielectric resonators.
8. L o o p - G a p R e s o n a t o r s One Loop - n Gap Resonator [8, pp. 515 - 517]
c
t
Figure 24. One Ioop-n gap resonator (n = 2 shown). Dimensions and features: a, loop; b, gaps; c, shield' d, inductive coupler; Z, resonator length; r, resonator radius; R, shield radius; t, gap separation; W, gap width [8, p. 516]. Reprinted with permission of Harcourt Brace and Company.
A coarse approximation for resonant frequency e WZ
C=
L=
tn
tZ o'n" r 2
Z
fo =
fo
is 1
2 rrv/LC
To account for the shield and flinging fields in the gap effects on resonant frequency, use the following equation:
r2
1
fo =
--
~/
nt
1~
1
1+
2rr
R 2-
(r + W) 2
7rW~/./,0 r
1 + 2.5
This equation is generally within 10% of measured resonant frequencies between 1 and 10 GHz. 6 = skin depth
Qo = 1
unloaded Q 1
=
Qo
1 +
Q1
Q2 '
84
Robert A. Strangeway
r[
where
01
[
=
1 + - - +r - 1.7
--
R 2 - (r
w R)(
1+
02
r2 ]
-1+ 8
+ W) 2
r2
R 2 -(r+W
r
)2] )2
105t
×
~0~,~,~(~ + ~ ~(~))
Mehdizadeh et al. [22] give corrections to fo and Qo for the finite length of the resonator.
Two Loop- One Gap Resonator, [9, pp. 1096- 1097] GAP
t
rl
ro A
L E ~ • SAMP LOOP
(3,® B-FIELD -,.-- EFIELD --.-- WALL CURRENTS
• L
" RETURNED FLUX LOOP
_1
I- W -I
AreA r:_~-----~ ~,,__-"~_ ,,, ; I:)
h
"
~ t l l l x xx I t (¢,'AI. Ill x x x x r ~ , , { x xxx ' r~.lllxxxxT' x xxx[? r/All.lxxxxl i
I~XXXi '
r,~litll ,
x
x xl
,
I Im I I
!
/
H-" I
~ tH i , r . ~ ,
,
l
l
~
.... B-FIELD X E FIELD J
! t L _ ~ . _ _ / j i /, ~, -_- --~--..~ - ,,
Figure 25. Two loop-one gap resonator. Reproduced with permission from Froncisz et ol. Rev. Sci. Instrum., Vol. 57, pp. 1095- 1099, 1986 and its publisher, American Institute of Physics.
A coarse approximation for resonant frequency fo is (c = speed of light):
( 2)
tl+ c
L
r°
r2
=
27rr o
7feW
3. Microwave Resonators
85
The corrections applied depend on the frequency of interest [9, p. 1096]. For a 35-GHz resonator, corrections of magnetic flux in the gap and fringing electric field were used:
L
t(1 + a) 2
C
2.5t )
2rr
E.rW 1 + ---~-
7rr 2 + 7ra2r 2 + ( 1 - a
+ a 2)- 3 -
where tW rrr2 + --~ a
=
tW 7rr ( +
7rr2 +
2
2
(1)Wt(1 5
7rr2a2 +
Oo=-g • 2 r r r o - t + (2rrr I - t ) a
2 +
a + a 2)
2 ) W ( 1 - a + a 2) -~
The reader is referred to Wood et al. [33] for the three loop-two gap resonator. Finally, it is noted that several other loop-gap resonator configurations exist; see, for example, Hyde and Froncisz [14].
9. Microstrip Resonators Microstrip Rectangular Resonators The approximate resonant frequency, fo [15, p. 152] is 2
L
2¢7
where e = permittivity of the microstrip dielectric /x = permeability of the microstrip dielectric Q =
~oo d lx
4Rw
[15, p. 153],
86
Robert A. Strangeway MICROSTRIP RESONATOR
-X
I-
F
20
:
L.--" MICROSTRIP RESONATOR
Ix Figure 26. Micros-triprectangular resonator dimensions [ 16, p. 947] [© 1974 IEEE].
where 1
R w = surface resistance = 1/ Y
q'rf/J, or
and tr = conductivity of the microstrip conductor. An accurate resonant frequency determination is given by Itoh [16], and an example of his results is reproduced in Figure 27. Wolff and Knoppik [32] give a method to calculate resonant frequency to within 1 to 2%. An improvement to 0.5% has been reported by Verma and Rostany [52]. Microstrip transmission line resonators are not covered here but should be treated as standard transmission line resonators, with appropriate end correction factors applied.
Microstrip Circular Resonators a - circular resonator radius.
87
3. Microwave Resonators
o._,,.o~= w = I cm
~' ~700
.-lcm w=2
c~
/~r-
•
w=--52~c~ 65 (r = 382
5001-
w
300-
i
6
~
,~
,lo
,~,
,~
,h
~lo
~
1
24
2/(cm)
Figure 27. Rectangular resonator resonant frequencies [I 6, p. 950] [© 1974 IEEE].
Note that all other variables are as defined under Microstrip Rectangular Resonators. An approximate resonant frequency and Q are 3.832 f o = 2,rc a v/ la, e
Q -
co o d / z
2R w
[15 pp. 151-152]. ,
Wolff and Knoppik [32] give a method to calculate f o to within approximately 1%. Bahl and Bhartia [1, pp. 79-82] review other calculation methods. Also see Ref. [50].
Microstrip Ring Resonators
W
\
w
I~-w--I
I-- w--I
I Figure 28. Closed and open ring resonators [30, p. 406] [© 1984 IEEE].
88
Robert A. Strangeway
Wu and Rosenbaum [34] present a mode chart for the (closed) ring resonator. Khilla [20] presents a method to calculate fo to within 1.5%. Tripathi and Wolff [30] present closed-form solutions for both the open and the closed ring resonators. Also see Ref. [50].
10. Fabry-Perot Resonators [6, pp. 337- 344]
d Figure 29. Ideal Fabry-Perot resonator [6, p. 338] [Robert Collin, Foundationsfor Microwave Engineering, © 1966 by McGraw- Hill, with permission of McGraw- Hill].
The ideal resonant frequency is fo = ( c n / 2 d ) , where n = 1,2,3,...
(mode)
c = speed of light. The ideal unloaded equation is Q = (,rrnZo/4Rw), where R w = surface resitance of the reflectors R w = ~'rrf~~r
Z o = intrinsic impedance of free space Z o = l.t/Ix___o
V
g o
Practical Fabry-Perot resonators must be coupled. Collin [6, pp. 339-344] describes one way of accomplishing this with circular holes in the endplates. The effects of the coupling holes on fo and Q are derived.
3. Microwave Resonators
89
Also, Fabry-Perot resonators often use spherical reflectors to ensure the confinement of the reflections [27, pp. 570-577]. See Refs. [35, 41, 42, 46] for equivalent circuits, coupling, and resonant frequency calculations of Fabry-Perot (open) resonators.
I I . YIG Resonators
Tv2
.o
. . • ;::'.. ...... ~'..:.. , . . . . :.
• ..'::'"'.~
h~:i
.... :./.!:i.~:~2~:
• o_ " "-. z.];-.:. ~(i~ -., . '_'.',.---~-_..~.:-~'r';,~
Figure 30. A three-stage YIG filter [From J. Helszajn, Passive and Active Microwave Circuits. Copyright © 1978 John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]
YIG = yttrium iron garnet (Y3Fe9012) fo = resonant frequency
F fo = 2,n- " ( H ° +- Ha),
[25,0.211]
90
Robert A. Strangeway
where MHz F = 0.221~
A/m
[1, p. 112; 25, p. 211]
H 0 = external biasing magnetic field intensity H a = inner anisotropic field of the YIG crystal. Note that the magnitude and sign depend on the orientation of the YIG crystal in the magnetic field. Osbrink [24] examines the Q of YIG spheres. YIG resonators are often used in filters and tuned oscillators. They are tuned linearly with respect to magnetic field bias. Mezak and Vendelin [43] review loop-coupled YIG resonator modeling.
12. Traveling-Wave Resonators Secondary microstrip
Primary m icrostrip
r
Input
r
Output
Figure 31. A traveling-wave resonator in a microstrip [15, p. 172]. [The figure was adapted from
Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company, reproduced with permission of the publisher.]
If the length of the secondary line is a multiple of whole wavelengths, a resonance response (absorption)will occur [12, pp. 201-202; 15, pp. 172-173]. Harvey also discusses how the traveling wave may be used to increase the fields' amplitudes in the resonant ring.
3. Microwave Resonators
91
References
[1] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design. New York: John Wiley & Sons, 1988. [2] W. Bazzy, publishers, The Microwave Engineers' Handbook and Buyers' Guide, 1963 ed. Brookline, MA: Horizon House, 1962. [3] R. N. Bracewell, Charts for resonant frequencies of cavities, Proc. IRE, Vol. 35, pp. 830-841, Aug. 1947. [4] A. B. Bronwell and R. E. Beam, Theory and Applications of Microwaves, pp. 377-383. New York: McGraw-Hill, 1947. [5] G. B. Brown, R. A. Sharpe, W. L. Hughes, and R. E. Post, Lines, Waves, and Antennas: The Transmission of Electric Energy, 2nd Ed. New York: John Wiley & Sons, 1973. [6] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1966. [7] C. W. Davidson, Transmission Lines for Communications with CAD Programs, 2nd Ed. New York: Wiley-Halsted Press, 1989. [8] W. Froncisz and J. S. Hyde, The loop-gap resonator: A new microwave lumped circuit ESR sample structure, J. Magn. Reson., Vol. 47, pp. 515-521, 1982. [9] W. Froncisz, T. Oles and J. S. Hyde, Q-band loop-gap resonator, Reu. Sci. Instrum., Vol. 57, pp. 1095-1099, 1986. [10] K. Fujisawa, General treatment of Klystron resonant cavities, IRE Trans. Microwave Theory Tech., Vol. MTT-6, pp. 344-358, Oct. 1958. [11] W. Geyi and W. Hongshi, Solution of the resonant frequencies of a microwave dielectric resonator using boundary element method, IEE Proc., Part H, Vol. 135, pp. 333-338, Oct. 1988. [12] A. F. Harvey, Microwave Engineering. London: Academic Press, 1963. [13] J. Helszajn, Passive and Active Microwave Circuits. New York: John Wiley & Sons, 1978. [14] J. S. Hyde and W. Froncisz, Loop-gap resonators, in Advanced EPR: Applications in Biology and Biochemistry (A. J. Hoff, ed.), pp. 277-306. Amsterdam: Elsevier, 1989. [15] T. K. Ishii, Microwave Engineering, 2nd Ed. San Diego: Harcourt Brace Jovanovich, 1989. [16] T. Itoh, Analysis of microstrip resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-22, pp. 946-952, Nov. 1974. [17] T. Itoh and R. S. Rudokas, New method for computing the resonant frequencies of dielectric resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 52-54, Jan. 1977. [18] D. Kajfez, Elementary functions procedure simplifies dielectric resonator's design, Microwave Syst. News, Vol. 12, pp. 133-140, June 1982 (© Argus Business). [19] D. Kajfez and P. Guillon, eds., Dielectric Resonators. Norwood, MA: Artech House, 1986. [20] A. M. Khilla, Ring and disk resonator CAD model, Microwave J., pp. 91-105, Nov. 1984. [21] R. W. P. King, H. R. Mimno, and A. H. Wing, Transmission Lines, Antennas, and Wave Guides. New York: Dover, 1965. [22] M. Mehdizadeh, T. K. Ishii, J. S. Hyde, and W. Froncisz, Loop-gap resonator: A lumped mode microwave resonant structure, IEEE Trans. Microwave Theory Tech., Vol. MTT31, pp. 1059-1064, 1983. [23] T. Moreno, Microwave Transmission Design Data. Norwood, MA: Artech House, 1989. [24] N. K. Osbrink, YIG-tuned oscillator fundamentals, Microwave Syst. Designer's Handb., Vol. 13, pp. 207-225, Nov. 1983. [25] E. Pehl, Microwave Technology. Dedham, MA: Artech House, 1985.
92
Robert A. Strangeway
[26] M. W. Pospieszalski, Cylindrical dielectric resonators and their applications in TEM line microwave circuits, IEEE Trans. Microwave Theory Tech., Vol. MTT-27, pp. 233-238, Mar. 1979). [27] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics. New York: John Wiley & Sons, 1965. [28] T. S. Saad, ed., Microwave Engineers' Handbook, Vols. 1 and 2. Dedham, MA: Artech House, 1971. [29] F. E. Terman, Resonant lines in radio circuits, Electr. Eng. (Am. Inst. Electr. Eng.), Vol. 53, pp. 1046-1053, July 1934. [30] V. K. Tripathi and I. Wolff, Perturbation analysis and design equations for open- and closed-ring microstrip resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-32, pp. 405-409, Apr. 1984. [31] I. G. Wilson, C. W. Schramm, and J. P. Kinzer, High Q resonant cavities for microwave testing, Bell Syst. Tech. Vol. 25, pp. 408-434, July 1946. [32] I. Wolff, and N. Knoppik, Rectangular and circular microstrip disk capacitors and resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-22, pp. 857-864, Oct. 1974. [33] R. L. Wood, W. Froncisz, and J. S. Hyde, The loop-gap resonator. II. Controlled return flux three-loop, two-gap microwave resonators for ENDOR and ESR spectroscopy, J. Magn. Reson., Vol. 58, pp. 243-253, 1984. [34] Y. S. Wu and F. J. Rosenbaum, Mode chart for microstrip ring resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-21, pp. 487-489, July 1973. [35] H. Cam, S. Toutain, P. Gelin, and G. Landrac, Study of a Fabry-Perot cavity in the microwave frequency range by the boundary element method, IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 298-304, Feb. 1992. [36] S.-W. Chen, and K. A. Zaki, Dielectric ring resonators loaded in waveguide and on substrate, IEEE Trans. Microwave Theory Tech., Vol. MTT-39, pp. 2069-2076, Dec. 1991. [37] D. Cros and P. Guillon, Author's reply, IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 1036-1038, May 1992. [38] D. Cros and P. Guillon, Whispering gallery dielectric resonator modes for W-band devices, IEEE Trans. Microwave Theory Tech., Vol. MTT-38, pp. 1667-1674, Nov. 1990. [39] W. K. Hui, and I. Wolff, Dielectric ring-gap resonator for application in MMIC's, IEEE Trans. Microwave Theory Tech., Vol. MTT-39, pp. 2061-2068, Nov. 1991. [40] E. N. Ivanov, D. G. Blair, and V. I. Kalinichev, Approximate approach to the design of shielded dielectric disk resonators with whispering-gallery modes, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 632-638, Apr. 1993. [41] T. Matsui, K. Araki, and M. Kiyokawa, Gaussian-beam open resonator with highly reflective circular coupling regions, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 1710-1714, Oct. 1993. [42] J. McCleary, M-y. Li, and K. Chang, Slot-fed higher order mode Fabry-Perot filters, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 1703-1709, Oct. 1993. [43] J. A. Mezak, and G. D. Vendelin, CAD design of YIG tuned oscillators, Microwave J., pp. 92-98, Dec. 1992. [44] R. K. Mongia, On the accuracy of approximate methods for analyzing cylindrical dielectric resonators, Microwave J., pp. 146-150, Oct. 1991. [45] R. K. Mongia, Resonant frequency of cylindrical dielectric resonator placed in an MIC environment, IEEE Trans. Microwave Theory Tech., Vol. MTI'-38, pp. 802-804, June 1990.
3. Microwave Resonators
93
[46] R. K. Mongia and R. K. Arora, Equivalent circuit parameters of an aperture coupled open resonator cavity, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 1245-1250, Aug. 1993. [47] R. K. Mongia and P. Bhartia, Easy resonant frequency computations for ring resonators, Microwave J., p. 105-108, Nov. 1992. [48] M. J. Niman, Comments on "Whispering galley dielectric resonator modes for W-band devices," IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 1035-1036, May 1992. [49] J.-S. Sun and C.-C. Wei, Iterative method provides accurate resonator analysis, Microwaves & RF, Vol. 30, pp. 70-79, Dec. 1991. [50] F. Tefiku and E. Yamashita, An efficient method for the determination of resonant frequencies of shielded circular disk and ring resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 343-346, Feb. 1993. [51] C. Vedrenne and J. Arnaud, Whispering-gallery modes of dielectric resonators, lEE Proc., Part H, Vol. 129, pp. 183-187, Aug. 1982. [52] A. K. Verma and Z. Rostamy, Resonant frequency of uncovered and covered rectangular microstrip patch using modified Wolff model, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 109-116, Jan. 1993. [53] W. Xi, W. R. Tinga, W. A. G. Voss, and B. Q. Tian, New results for coaxial re-entrant cavity with partially dielectric filled gap, IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 747-753, Apr. 1992.
3 Microwave Resonators Robert A. Strangeway
I. Resonator Basics One-Port Reflection Resonator
G
C f
~
Figure I. Parallel resonant circuit.
1
~°°= CLC
= resonant radian frequency
tooC QU --
G
- u n l o a d e d quality factor (.JO o
Y = G 1 + jQu t°o
Handbook of Microwave Technology, Volume I
= circuit a d m i t t a n c e . CO
61
Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
Robert A. Strangeway
62
T-Line 1 L
G
C f ~
Go
Figure 2. Losslesstransmission line terminated in a resonant circuit.
If to = to o + Ato, then in the vicinity of resonance, Ato << too, the approximate admittance expression is Y = G(1 + jQu 2 Ato/too).
G O = transmission line characteristic conductance /3 =
60 G
= coupling factor
(/3 > 1, o v e r c o u p l e d ; / 3 = 1, critically c o u p l e d ; / 3 < 1, u n d e r c o u p l e d ) 2Ato (/3 - 1) - J Q u ~ too
F =
2Ato ( / 3 + 1) + J a u ~
= reflection coefficient
too
u n l o a d e d Q = Qu external Q = Qe
tooC G too C
60 ~o C
loadedQ=QL=
1
1
QL OH
ae
+
(Go+G)
1
au
=l+fl
au =/3 ae IFol = reflection coefficient m a g n i t u d e at to o
(1 -trot) /3 = (1 +
IFol)
(/3 < 1)
63
3. Microwave Resonators or
(1 +
IFul =
IUol)
r e f l e c t i o n c o e f f i c i e n t m a g n i t u d e for
IFul- ( (/~ - 1)2+ 1 (/~ + 1) ~ + 1 IFel = r e f l e c t i o n c o e f f i c i e n t
QL measurement
m a g n i t u d e for
Qe m e a s u r e m e n t
r e f l e c t i o n c o e f f i c i e n t m a g n i t u d e for
OL m e a s u r e m e n t
/ ( ~ __ 1) 2 + ~2 Irel-- (/3 + 1) 2 "q-~2 IFLI =
]~L]-- ¢
(]~°]22+ 1)
Two-Port Transmission Resonator [23, pp. 240-241]
] T-Line G°l
G
.
I
C
T-Line 2 G°2
"~
Figure 3. Two-port transmission resonatorcoupled to two transmission lines.
oJoC
Ou =
unloaded resonator Q =
ael =
external Q of input circuit =
Oe2 =
e x t e r n a l Q o f o u t p u t circuit -
QL, =
loaded Q without port 2
G
~o C
Q1 oJoC
02
64
Robert A. Strangeway
QL1
O9oC G+Go I
QL2--
loaded Q without port 1
QL2 O,~l,~ =
auQel Qu+Qel
090C
QuQe2
G+Go 2
Qu+Qe2
l o a d e d Q with b o t h p o r t s
oJoC
QLI,~
G + Gol -+-Go2 1
QL1,2
+
1
-F~
Qel
Qu
1
Qe2
/31 = p o r t 1 coupling factor =
Go 1 G
/3 2 = p o r t 2 coupling factor =
=
Qu Qel
Go2 Qu = G Qe2
QU
QLI,2
=
+ ~1 q" ~2
1
S 1 = s t a n d i n g - w a v e ratio of p o r t 1 at r e s o n a n c e with p o r t 2 m a t c h e d
131 3'1-"
1+132
$1 = 3'1 if 3'1 > 1; o t h e r w i s e S 1 = 1/3'1 S 2 = s t a n d i n g - w a v e ratio of p o r t 2 at r e s o n a n c e with p o r t 1 m a t c h e d ~2 Y2
1+/31
$2 -- 72 if 72 >
1;
otherwise S 2 =
1/72
[To[ 2 = p o w e r t r a n s m i s s i o n coefficient at oJo
ITol2 [12, pp. 200-201].
.---
471 (1 + 3'1) 2
,
~1 -- 3'1 ~1
.---
4131132 (1 + fll + f12) 2
3. Microwave Resonators
65
Skin Depth,
w h e r e / z is the permeability and or is the conductivity (SI units for all variables).
2. Two-Wire Line Resonators Resonance occurs when L = h o / 4 (quarter wave) or h o / 2 (half wave) and the line is properly terminated, usually with a short termination (an open termination tends to radiate). Radiation and termination reactances [21, pp. 286-288] will change fo. a = transmission line attenuation constant /3 = transmission line phase constant. The unloaded Q = Qu for a transmission line with low losses is Qu = (/3/ 2 a ) [7, pp. 77-79]. An approximate Qu for a two-wire line resonator is Qu = (Ixd2°~o/8Rwa) [15, p. 150], where /
R w = surface resistance = ~/ Y
T/'f~ or
or = conductivity /x = permeability Loaded Q must incorporate termination losses [21, pp. 288-293]. Terman ([29, p. 1049) determined the Q for nonradiating two-wire copper lines as Q = 0.08877rfbJ, where b = d in Figure 4 and J is read from the graph in Figure 5.
K
L
I\
/ Short or
d
Open
Figure 4. Basic two-wire line resonator.
66
Robert A. Strangeway 1.0
- ~ ~
0.9
%%
//
0.8 03 0.6 zO.5 0.4
0.3
-
0.2 0.1 0
2
I
5
I0
b/.~
20
50
I00
Figure 5. Terman's J factor (ignore other factors) [29, p. 1049] [© 1934AIEE (now IEEE)].
3. Basic Coaxial Resonators [23, pp. 2 2 5 - 238]
I
I
I
I
I
2zo
b Figure 6. Coaxial resonator [23, p. 227]. Reproducedwith permission of Artech House, Inc.
67
3. Microwave Resonators
TEM mode resonant frequencies are as follows: 2Z o = nAo/2 , where n = 1, 2, 3 , . . . and Ao is the TEM wavelength. (Note. Here, Z o is the half cavity length, not characteristic impendance.) Note that significant losses perturb fo. Q is the unloaded quality factor; when n = 1,
1 b
AO
l+-
2Zo
4-~,
[23, p. 226],
a
In -
a
where 6 is the skin depth (see Section 1). Be warned that coaxial line resonators can support non-TEM modes. An approximate formula for the higher mode cutoff frequency, fc, in coaxial transmission lines is [5, p. 259] ¢
fc
7r(a + b ) '
where c is the free-space velocity of light, ~ is the permeability of the medium, and e is the permittivity of the medium. Foreshortened Coaxial Line Resonators (Reentrant Coaxial Cavity Resonators) 2r °
.]'
d
½
~'--
2 rt
"7"-I
Figure 7. Foreshortened coaxial line resonator dimensions [I 5, p. 143]. [The figure was adapted from
Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.]
The resonant frequency is fo = (1/27rx/Lc), where /xh r1 L= In 2re ro 2
C=e°
rrr____& d _ 4ro In
(
2d e C h 2 + (r 1 - ro) 2
)]
[10, p. 3491
68
Robert A. Strangeway
and e is the base of natural logarithms. Resonant frequency design charts are also given in Moreno [23, pp. 230-238] and are accurate to within a few percent for gaps that are not too large. For these resonators, the approximate Q is r 1
ln-Q6
2h
,t °
A°
ro
rI 21nm +h ro
1 1 ] m+-ro
[23, p. 229].
r 1
Xi et al. [53] analyze a coaxial reentrant cavity with a dielectric in the gap.
4. R e c t a n g u l a r
t
Waveguide
X,-
Cavities
2
1A±)'. c÷)', c÷ Q
Q~
MODE
TE~mn
TEOmn
(p= + q2)(p2 + q2 + r~)s/2 abe • 4 ,c[p=r 2 + (i)2 + q2)2] + bc (q2r~ + (1=3+ q2)2] + obr ~ (pt + Cl~)
(q2 + r2) s/2
abc T"
q2c (b + 2a) + r~b(c + 2a)
TE¢On
obc (p2 + ,2) ~/= 2 " p2c(a + 2b) + r2a(c + 2b)
TM~..
.bc. (p2 + q=) (!~ + q2+ ,2),/2 4
TM¢~o
i ) l b ( a + c) +q2a (b + c)
obc
(p2 + q2) s/2
2 " p=b(a + 2c) + q2a(b + 2c)
p=
~, a
q.
m b
r--
ii
h e = Resonant Free-Space Wavelength 6 -- Skin Depth Figure 8. Resonant wavelength and Q [28, Vol. I, p. 180]. Reproduced with permission of Artech House, Inc. Note that Bazzy [2] also provides many of the figures in this and the next chapter.
69
3. Microwave Resonators
E:.
l-7~
131,311 I-6
-.4 ~TM 130, T M 3 1 0
~r~
T(O31.TE3OI
221
"rico3
-_
?.2
1.5~
6T[OI2.1T.I02 ~1-5
TM220 1.4-121.211
1'(O21 . T ( 2 0 1 1.3 "-"
m2.o 1.2 - -
D2.1 m "r[Oll. T[IOI
I.I
m2-2
"I "I'M 1 2 0 , T M 2 1 0 --
B
m2.3 i 1.0
el
B2.4
R
EXPLANATION 0-9
-_
L
---2.5
_ 0.8
--
0.7
~
0-6
0.5
B2.6
TMIIO
--2-7
-
o
T(
O
T [ & T M Modes.
Modes
only.
0
T M Mode~ only.
--2-.
--2e
Figure 9. Mode lattice for square waveguide resonators [3, p 834] [© 1947 IRE (now IEEE)]
70
Robe~ A. Strangeway
.--, ..
I
I
¢!" .... ~ I f.;'-q!
A,I~I / ! II I .~a~.-F~-4
I
.~4-
o
~
IXi ~.24; i !I III ~ , - - ,41,,- - ,~
r 4_
If ~ - 4 - - - - - ,
TMlll
It I/
v
TM~12 (see note)
IY
I TMl13 / (me note) /z
-X
/[ ,-.-_'-_-~
. r_-_2", [,r~.;__IvV 1/':..~_:) 1/' TM121
TM122 (see note)
TM123 (see note)
TM131 (see note)
TM132 (see note)
TMI~ (see note)
Figure10. TM mode field pattern sketches. Solid lines, E; dashed lines, H [I 5, p. 127]. [The figures are adapted from Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
3. Microwave Resonators
71
I
,.~~~:s~,t /
VtS"_----.-----~'V TEIm
TEIu
TEI~
TE lm (see note)
TErn
TEsu
(see note)
(see no~)
TErn (see note)
TE m (see note)
~
x
A, ~ - - - - ' : "IA
/ ~:2-",~/I .
-
-
II / ~.--.~.--Jlll
V ~'-"--"Y
TE~a
Figure II. TE mode field pattern sketches, Solid lines, E; dashed lines, H [15, p. 125]. [The figures are adapted from Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
72
Robert A.
Strangeway
% +0.01
--100
--
- 90
-
- 80
- 70 I.-
z i
•
U
-60
| v >-
I-
z. u U
o'
Og o.
~
5o ~ i
_>
EXAMPLE -0.03 - . -
%
%
+0.01
.100
_
..40
-
-30
i m
.90 0 80
-0.04
-
-0.01
•70
-0.02
60 50
"
-0.03
-
-0.04 -0.05
-0.05
-0.06
40
30 2O 10 0
NOMOGRAPH IS NORMAUZED RELATIVE TO 25"C AND 60% REL. HUMIDITY. -0.06
-20
-10
0
Figure 12. Humidity-cavity resonant frequency nomograph [28, Vol. I, p. 185]. Reproduced with permission of Artech House, Inc. [MIT RacI Lab Series, Techniqueof Microwave Measurement, Vol. I I 1944, McGraw-Hill; reproduced with permission of McGraw-Hill.]
73
3. Microwave Resonators
5. C i r c u l a r
Cylindrical
Waveguide
Cavities
WAVELENGTH AND Q FOR TEem. OR TMCmn MODES
-tt
~,.=
2
L
I
÷(:),
~,o Resonant Wavelength in Free Space Xc. mth root of J'¢(X) = O for the TE modes Xs.. mth root of J {[ X) = O for the TM modes ,,
,,
MODE
TEcmn
I - (XCm)=]J [X~m+ p'r2] 31'
Xcm
TMcmn
TM~mo
VX2¢m + par:
(n > o)
2r ( l + r )
Xcm (n
:r(2 + r)
5 = Skin Depth
r = D
L
-- o)
p = nf
2
Figure 13. Resonant wavelength and Q [28, Vol. I, p. 180]. Reproduced with permission of Artech House, Inc. Note that the humidity-cavity resonant frequency homograph appears in Figure 12.
74
Robert A. Strangeway
TH~ FIaST TWENTY-FIVE ZEROS OF J . ( x ) AND J . ' ( x ) No.
Value
Mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1.841 2.405 3.054 3.832 3.832 4.201 5.136 5.318 5.332 5.520 6.380 6.416 6.706 7.016 7.016 7.501 7.588 8.016 8.417 8.536 8.578 8. 654 8.771 9.283 9.647
TE~ TMol TE2x TMll TEol TEsx TM21 TE4x TEl2 TMo2 TM31 TE61 TE22 TM12 TEo2 TE61 TM~t TE82 TM~2 TExs TEn TMo~ TM51 TE~2 TE81
Figure 14. Bessel function zeros [23, p. 222]. Reproduced with permission of Artech House, Inc. Note that Saad [28, Vol. 2, pp. 192-195] has a more extensive listing.
75
3. Hicrowave Resonators
mO
I TMO2--~ r-
1'9 ~-TM022 _~TEI22 -~TF..412 -~-TM021
n
0.5
+TE3~3 -~T[,OI3, TMII3
TMQ~O
=I~TF-121 [411
+Te:2~3
I-0
--~-.TMOI3 -~)-TF..113
4-TE312 -~TF..OI2 ,TM II 2
T(12~ --1"7 T(44
--
"FTM212
1'8
TM211
T M 2 1 - - r"
-~TM023 ,,~ -~-TEI23LT[413 m -I)-TM213
TM210
+T.o,2 +Tu,2
1-6 +TE311
--
1-5 TF.OII, TMII
~
1.4
D
D
L
~
TE,21t
TE3I-- - - 1.3
+TMOI I TMIIO 1"2
1T.OI ,'I'M I1--~
m2.0 +TE III
I X PL AN ATION
~I'I /
[
I'O TE21 m -- O.9
TMOI"
--
0"8 TMOIO
-- 0 " 7 T£11m -- 0 " 6 --'O'5 - - 2-S
Figure 15. Mode lattice for cylindrical resonators [3, p. 832]. [© 1947 IRE (now IEEE)].
76
Robert A. Strangeway
TM 022 TM 021 3O
TE 122TM 212 28 TE 412, 26
24
ii~ii?i~ 22
20
~
16
u
14
z o
12
0
D
×
~..
e
10
i-~ii.i.!i. M O D E CHART FOR RIGHT CIRCULAR C Y U N D E R
0
I
2
3
4
S
6
7
8
9
(D/L)'
Figure 16. Mode chart for cylindrical resonators [28, Vol. I, p. 181]. Reproduced with permission of Artech House, Inc.
O4
_
t,')
..j
~.t'-
=7 ClOHI3 ~1 9 N I 7 d n 0 3
y
)
rn
w
rn
~I
{3
~ ;?J
~
w
w
m oJ
CD
~r
~< "-
w
-< ~ 3:
N
N
r
JC:
-J
c:o~ ,(
~
~,
9-
I
~l
~< •
~
"<~
"~
~
~1:::11~10 ~ V 7 N : ) ~ I D
~:
'<"
--
-
0
N
d66
~
C
d
_
5
-coo m -
6
~
d
O 0 ~ -
~
r"
o o o
_ ( ~ w w w SINVISNOD
0
m
~
5
(~
d
~"
:3
m "E
w
<
om Z~(M
I~I
~z~
N
toO_
~_5 o
_
m 0
¢)
._c
x ¢)
>--
~-J ~'-~
@m
~'
~
~
c
"~
of-
~g 0
~
~.
~
-
.gm
u
-
}j
E °
,,
I--
Z
IE
o >wz ~I0 w r--
.o
0--
I0
z
U
w
~
I--~
_c:
~.o
~Z
EL
o
~ +-.
~
L2~ .j
w
=
Z
_
~
Ow
~
~w
,,,,,..
uJ
L.
.,
<._
~_ _E
~.
p.~
~
~
o ~__
~:)
z
/J
TMulmode
TMoxI mode
E M . . . . .
TMo~
TMom
~
TMo~
m
I
I TMI~
TMm
TMm
( m note)
( m nora)
%
I
' 1 4&_.
g 5 ~J-.'~-',.' .,:)_;)
TM212 (seenote)
TM:m (seenote)
TM~m (seenote)
Figure 18. TM mode field pattern sketches [15, pp. 134-135]. [The figures are adapted from
Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
I I t
I
t
,,
i
i ~ \
,
! /
.
I1,
I A
~
/t'
/t'
It It
h i
Jl ',
"~-E
(a) TEosxmode
(b) TEsss mode
TEos~
TErns
TEoaz (see note)
TEss2
TErn (see note)
TE~s (see note)
m
TE212
TEmx
TEzsz
(see note)
(see note)
Figure 19. TE mode field pattern sketches [15, pp. 130-131]. [The figures are adapted from Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company reproduced with permission of the publisher.] Note: this mode is a repetition of the basic mode. Some field lines have been omitted for clarity.
79
80
Robert A. Strangeway
6. Spherical Cavities [4, p. 380]
E
H
Figure 20. Field patterns for the lowest order modes in spherical cavities. (a) TE i i0 mode. (b) TM i i0 mode. [From A. B. Bromwell and R. E. Beam, Theory and Applications of Microwaves, © 1947 McGraw-Hill with permission of McGraw-Hill.]
a
=
spherical cavity radius
wavelength
h r --
resonant
/~r =
1.40a for TEll m modes 2.29a for T M l l m modes.
~'r --
7. Dielectric Resonators
l
Z
Figure 21. Dielectric resonator dimensions and magnetic field for the TE0i 8 mode [19, p. 2]. Reproduced with permission of Artech House, Inc.
3. Microwave Resonators
81
The approximate resonant frequency, free space in the TE01 ~ mode is a ~34
fo-
fo, of a
[ aL- +3"45 1
dielectric resonator (DR) in
(GHz)
[19, p. 3],
where a = radius of the DR in millimeters L = length of the DR in millimeters er
relative permittivity of the DR.
=
This equation is accurate to approximately 2% under the conditions 0.5 < (a/L) < 2 and 30 < e r < 50. More accurate and mathematically intensive calculation procedures which account for the environment around the resonator, such as a microstrip substrate and metal shields, exist (see, for example, [11, 17, 19, 26]). An example of a mode chart, from Pospieszalski [26, p. 235], is shown in Figure 22. Later literature improves the accuracy, ease of calculations, or both and includes references to several approximate and rigorous techniques [44, 45, 49]. The determination of the length of a DR for a given diameter and frequency can be calculated to within a few percent using Kajfez's procedure ([18]; also shown in [19, pp. 148-149]) for the Itoh and Rudokas
(fo'D)2"erl
i
[GHz cm]2! ~0
4.103
"
L1 SI='E
TE01 1 mode ~
2.1o3
er I> 500 er =10
.~j~" . / ~ I"~
.....-- ..._--'-
......
S1= .
...~
j
. . f
. . . .
:
s~=o.4 . . . . .
1o103
o
0.5 1.0
i
o
i
o
i
i
i
i
o
i
i
i
i
i~
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure 22. A DR mode chart [© 1979 IEEE].
112
Robert A. Strangeway
I -
o
L
i ....
. JI
.
Er
E2 .
.
.
.
.
.
1
Ib ID Ill
T L2
t ......... mm
--
Figure 23. DR geometry and dimensions, e 2, is the substrate relative permittivity [18, p. 133] [© by Argus Business, reprinted with permission].
model [17]. Refer to Figure 23 for the geometry and variable definitions. Note the metallic cover at distance L 1 above the resonator. The straightforward calculations are listed below. Specify fo. Select D: 5.4
5.4
ko~r
< D < ~koCe2 ,
where
k o = free-space propagation constant = toV/~oe o Yo=
ko- ~
(e r -
1)-2.4052
2
Yo
h-~
.405+
(
2.43 ) 2.405 1 + ~ + 0.291y o
Yo fl
=
Ck oF, 2 r
--
h2
al= Ch2-k 2 Ot 2
L
=
=
Ch 2
-
_
2 2 koe
/3 t a n - 1 ~/3 coth al L 1 + tan - 1 ~a~2 coth a2L 2 -
.
83
3. Microwave Resonators
See [36, 39, 47] for dielectric ring resonators. See [37, 38, 40, 48, 51] for whispering gallery mode dielectric resonators.
8. L o o p - G a p R e s o n a t o r s One Loop - n Gap Resonator [8, pp. 515 - 517]
c
t
Figure 24. One Ioop-n gap resonator (n = 2 shown). Dimensions and features: a, loop; b, gaps; c, shield' d, inductive coupler; Z, resonator length; r, resonator radius; R, shield radius; t, gap separation; W, gap width [8, p. 516]. Reprinted with permission of Harcourt Brace and Company.
A coarse approximation for resonant frequency e WZ
C=
L=
tn
tZ o'n" r 2
Z
fo =
fo
is 1
2 rrv/LC
To account for the shield and flinging fields in the gap effects on resonant frequency, use the following equation:
r2
1
fo =
--
~/
nt
1~
1
1+
2rr
R 2-
(r + W) 2
7rW~/./,0 r
1 + 2.5
This equation is generally within 10% of measured resonant frequencies between 1 and 10 GHz. 6 = skin depth
Qo = 1
unloaded Q 1
=
Qo
1 +
Q1
Q2 '
84
Robert A. Strangeway
r[
where
01
[
=
1 + - - +r - 1.7
--
R 2 - (r
w R)(
1+
02
r2 ]
-1+ 8
+ W) 2
r2
R 2 -(r+W
r
)2] )2
105t
×
~0~,~,~(~ + ~ ~(~))
Mehdizadeh et al. [22] give corrections to fo and Qo for the finite length of the resonator.
Two Loop- One Gap Resonator, [9, pp. 1096- 1097] GAP
t
rl
ro A
L E ~ • SAMP LOOP
(3,® B-FIELD -,.-- EFIELD --.-- WALL CURRENTS
• L
" RETURNED FLUX LOOP
_1
I- W -I
AreA r:_~-----~ ~,,__-"~_ ,,, ; I:)
h
"
~ t l l l x xx I t (¢,'AI. Ill x x x x r ~ , , { x xxx ' r~.lllxxxxT' x xxx[? r/All.lxxxxl i
I~XXXi '
r,~litll ,
x
x xl
,
I Im I I
!
/
H-" I
~ tH i , r . ~ ,
,
l
l
~
.... B-FIELD X E FIELD J
! t L _ ~ . _ _ / j i /, ~, -_- --~--..~ - ,,
Figure 25. Two loop-one gap resonator. Reproduced with permission from Froncisz et ol. Rev. Sci. Instrum., Vol. 57, pp. 1095- 1099, 1986 and its publisher, American Institute of Physics.
A coarse approximation for resonant frequency fo is (c = speed of light):
( 2)
tl+ c
L
r°
r2
=
27rr o
7feW
3. Microwave Resonators
85
The corrections applied depend on the frequency of interest [9, p. 1096]. For a 35-GHz resonator, corrections of magnetic flux in the gap and fringing electric field were used:
L
t(1 + a) 2
C
2.5t )
2rr
E.rW 1 + ---~-
7rr 2 + 7ra2r 2 + ( 1 - a
+ a 2)- 3 -
where tW rrr2 + --~ a
=
tW 7rr ( +
7rr2 +
2
2
(1)Wt(1 5
7rr2a2 +
Oo=-g • 2 r r r o - t + (2rrr I - t ) a
2 +
a + a 2)
2 ) W ( 1 - a + a 2) -~
The reader is referred to Wood et al. [33] for the three loop-two gap resonator. Finally, it is noted that several other loop-gap resonator configurations exist; see, for example, Hyde and Froncisz [14].
9. Microstrip Resonators Microstrip Rectangular Resonators The approximate resonant frequency, fo [15, p. 152] is 2
L
2¢7
where e = permittivity of the microstrip dielectric /x = permeability of the microstrip dielectric Q =
~oo d lx
4Rw
[15, p. 153],
86
Robert A. Strangeway MICROSTRIP RESONATOR
-X
I-
F
20
:
L.--" MICROSTRIP RESONATOR
Ix Figure 26. Micros-triprectangular resonator dimensions [ 16, p. 947] [© 1974 IEEE].
where 1
R w = surface resistance = 1/ Y
q'rf/J, or
and tr = conductivity of the microstrip conductor. An accurate resonant frequency determination is given by Itoh [16], and an example of his results is reproduced in Figure 27. Wolff and Knoppik [32] give a method to calculate resonant frequency to within 1 to 2%. An improvement to 0.5% has been reported by Verma and Rostany [52]. Microstrip transmission line resonators are not covered here but should be treated as standard transmission line resonators, with appropriate end correction factors applied.
Microstrip Circular Resonators a - circular resonator radius.
87
3. Microwave Resonators
o._,,.o~= w = I cm
~' ~700
.-lcm w=2
c~
/~r-
•
w=--52~c~ 65 (r = 382
5001-
w
300-
i
6
~
,~
,lo
,~,
,~
,h
~lo
~
1
24
2/(cm)
Figure 27. Rectangular resonator resonant frequencies [I 6, p. 950] [© 1974 IEEE].
Note that all other variables are as defined under Microstrip Rectangular Resonators. An approximate resonant frequency and Q are 3.832 f o = 2,rc a v/ la, e
Q -
co o d / z
2R w
[15 pp. 151-152]. ,
Wolff and Knoppik [32] give a method to calculate f o to within approximately 1%. Bahl and Bhartia [1, pp. 79-82] review other calculation methods. Also see Ref. [50].
Microstrip Ring Resonators
W
\
w
I~-w--I
I-- w--I
I Figure 28. Closed and open ring resonators [30, p. 406] [© 1984 IEEE].
88
Robert A. Strangeway
Wu and Rosenbaum [34] present a mode chart for the (closed) ring resonator. Khilla [20] presents a method to calculate fo to within 1.5%. Tripathi and Wolff [30] present closed-form solutions for both the open and the closed ring resonators. Also see Ref. [50].
10. Fabry-Perot Resonators [6, pp. 337- 344]
d Figure 29. Ideal Fabry-Perot resonator [6, p. 338] [Robert Collin, Foundationsfor Microwave Engineering, © 1966 by McGraw- Hill, with permission of McGraw- Hill].
The ideal resonant frequency is fo = ( c n / 2 d ) , where n = 1,2,3,...
(mode)
c = speed of light. The ideal unloaded equation is Q = (,rrnZo/4Rw), where R w = surface resitance of the reflectors R w = ~'rrf~~r
Z o = intrinsic impedance of free space Z o = l.t/Ix___o
V
g o
Practical Fabry-Perot resonators must be coupled. Collin [6, pp. 339-344] describes one way of accomplishing this with circular holes in the endplates. The effects of the coupling holes on fo and Q are derived.
3. Microwave Resonators
89
Also, Fabry-Perot resonators often use spherical reflectors to ensure the confinement of the reflections [27, pp. 570-577]. See Refs. [35, 41, 42, 46] for equivalent circuits, coupling, and resonant frequency calculations of Fabry-Perot (open) resonators.
I I . YIG Resonators
Tv2
.o
. . • ;::'.. ...... ~'..:.. , . . . . :.
• ..'::'"'.~
h~:i
.... :./.!:i.~:~2~:
• o_ " "-. z.];-.:. ~(i~ -., . '_'.',.---~-_..~.:-~'r';,~
Figure 30. A three-stage YIG filter [From J. Helszajn, Passive and Active Microwave Circuits. Copyright © 1978 John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]
YIG = yttrium iron garnet (Y3Fe9012) fo = resonant frequency
F fo = 2,n- " ( H ° +- Ha),
[25,0.211]
90
Robert A. Strangeway
where MHz F = 0.221~
A/m
[1, p. 112; 25, p. 211]
H 0 = external biasing magnetic field intensity H a = inner anisotropic field of the YIG crystal. Note that the magnitude and sign depend on the orientation of the YIG crystal in the magnetic field. Osbrink [24] examines the Q of YIG spheres. YIG resonators are often used in filters and tuned oscillators. They are tuned linearly with respect to magnetic field bias. Mezak and Vendelin [43] review loop-coupled YIG resonator modeling.
12. Traveling-Wave Resonators Secondary microstrip
Primary m icrostrip
r
Input
r
Output
Figure 31. A traveling-wave resonator in a microstrip [15, p. 172]. [The figure was adapted from
Microwave Engineering, 2nd ed., by T. Koryu Ishii, © 1989 by Harcourt Brace and Company, reproduced with permission of the publisher.]
If the length of the secondary line is a multiple of whole wavelengths, a resonance response (absorption)will occur [12, pp. 201-202; 15, pp. 172-173]. Harvey also discusses how the traveling wave may be used to increase the fields' amplitudes in the resonant ring.
3. Microwave Resonators
91
References
[1] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design. New York: John Wiley & Sons, 1988. [2] W. Bazzy, publishers, The Microwave Engineers' Handbook and Buyers' Guide, 1963 ed. Brookline, MA: Horizon House, 1962. [3] R. N. Bracewell, Charts for resonant frequencies of cavities, Proc. IRE, Vol. 35, pp. 830-841, Aug. 1947. [4] A. B. Bronwell and R. E. Beam, Theory and Applications of Microwaves, pp. 377-383. New York: McGraw-Hill, 1947. [5] G. B. Brown, R. A. Sharpe, W. L. Hughes, and R. E. Post, Lines, Waves, and Antennas: The Transmission of Electric Energy, 2nd Ed. New York: John Wiley & Sons, 1973. [6] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1966. [7] C. W. Davidson, Transmission Lines for Communications with CAD Programs, 2nd Ed. New York: Wiley-Halsted Press, 1989. [8] W. Froncisz and J. S. Hyde, The loop-gap resonator: A new microwave lumped circuit ESR sample structure, J. Magn. Reson., Vol. 47, pp. 515-521, 1982. [9] W. Froncisz, T. Oles and J. S. Hyde, Q-band loop-gap resonator, Reu. Sci. Instrum., Vol. 57, pp. 1095-1099, 1986. [10] K. Fujisawa, General treatment of Klystron resonant cavities, IRE Trans. Microwave Theory Tech., Vol. MTT-6, pp. 344-358, Oct. 1958. [11] W. Geyi and W. Hongshi, Solution of the resonant frequencies of a microwave dielectric resonator using boundary element method, IEE Proc., Part H, Vol. 135, pp. 333-338, Oct. 1988. [12] A. F. Harvey, Microwave Engineering. London: Academic Press, 1963. [13] J. Helszajn, Passive and Active Microwave Circuits. New York: John Wiley & Sons, 1978. [14] J. S. Hyde and W. Froncisz, Loop-gap resonators, in Advanced EPR: Applications in Biology and Biochemistry (A. J. Hoff, ed.), pp. 277-306. Amsterdam: Elsevier, 1989. [15] T. K. Ishii, Microwave Engineering, 2nd Ed. San Diego: Harcourt Brace Jovanovich, 1989. [16] T. Itoh, Analysis of microstrip resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-22, pp. 946-952, Nov. 1974. [17] T. Itoh and R. S. Rudokas, New method for computing the resonant frequencies of dielectric resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 52-54, Jan. 1977. [18] D. Kajfez, Elementary functions procedure simplifies dielectric resonator's design, Microwave Syst. News, Vol. 12, pp. 133-140, June 1982 (© Argus Business). [19] D. Kajfez and P. Guillon, eds., Dielectric Resonators. Norwood, MA: Artech House, 1986. [20] A. M. Khilla, Ring and disk resonator CAD model, Microwave J., pp. 91-105, Nov. 1984. [21] R. W. P. King, H. R. Mimno, and A. H. Wing, Transmission Lines, Antennas, and Wave Guides. New York: Dover, 1965. [22] M. Mehdizadeh, T. K. Ishii, J. S. Hyde, and W. Froncisz, Loop-gap resonator: A lumped mode microwave resonant structure, IEEE Trans. Microwave Theory Tech., Vol. MTT31, pp. 1059-1064, 1983. [23] T. Moreno, Microwave Transmission Design Data. Norwood, MA: Artech House, 1989. [24] N. K. Osbrink, YIG-tuned oscillator fundamentals, Microwave Syst. Designer's Handb., Vol. 13, pp. 207-225, Nov. 1983. [25] E. Pehl, Microwave Technology. Dedham, MA: Artech House, 1985.
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Robert A. Strangeway
[26] M. W. Pospieszalski, Cylindrical dielectric resonators and their applications in TEM line microwave circuits, IEEE Trans. Microwave Theory Tech., Vol. MTT-27, pp. 233-238, Mar. 1979). [27] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics. New York: John Wiley & Sons, 1965. [28] T. S. Saad, ed., Microwave Engineers' Handbook, Vols. 1 and 2. Dedham, MA: Artech House, 1971. [29] F. E. Terman, Resonant lines in radio circuits, Electr. Eng. (Am. Inst. Electr. Eng.), Vol. 53, pp. 1046-1053, July 1934. [30] V. K. Tripathi and I. Wolff, Perturbation analysis and design equations for open- and closed-ring microstrip resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-32, pp. 405-409, Apr. 1984. [31] I. G. Wilson, C. W. Schramm, and J. P. Kinzer, High Q resonant cavities for microwave testing, Bell Syst. Tech. Vol. 25, pp. 408-434, July 1946. [32] I. Wolff, and N. Knoppik, Rectangular and circular microstrip disk capacitors and resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-22, pp. 857-864, Oct. 1974. [33] R. L. Wood, W. Froncisz, and J. S. Hyde, The loop-gap resonator. II. Controlled return flux three-loop, two-gap microwave resonators for ENDOR and ESR spectroscopy, J. Magn. Reson., Vol. 58, pp. 243-253, 1984. [34] Y. S. Wu and F. J. Rosenbaum, Mode chart for microstrip ring resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-21, pp. 487-489, July 1973. [35] H. Cam, S. Toutain, P. Gelin, and G. Landrac, Study of a Fabry-Perot cavity in the microwave frequency range by the boundary element method, IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 298-304, Feb. 1992. [36] S.-W. Chen, and K. A. Zaki, Dielectric ring resonators loaded in waveguide and on substrate, IEEE Trans. Microwave Theory Tech., Vol. MTT-39, pp. 2069-2076, Dec. 1991. [37] D. Cros and P. Guillon, Author's reply, IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 1036-1038, May 1992. [38] D. Cros and P. Guillon, Whispering gallery dielectric resonator modes for W-band devices, IEEE Trans. Microwave Theory Tech., Vol. MTT-38, pp. 1667-1674, Nov. 1990. [39] W. K. Hui, and I. Wolff, Dielectric ring-gap resonator for application in MMIC's, IEEE Trans. Microwave Theory Tech., Vol. MTT-39, pp. 2061-2068, Nov. 1991. [40] E. N. Ivanov, D. G. Blair, and V. I. Kalinichev, Approximate approach to the design of shielded dielectric disk resonators with whispering-gallery modes, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 632-638, Apr. 1993. [41] T. Matsui, K. Araki, and M. Kiyokawa, Gaussian-beam open resonator with highly reflective circular coupling regions, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 1710-1714, Oct. 1993. [42] J. McCleary, M-y. Li, and K. Chang, Slot-fed higher order mode Fabry-Perot filters, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 1703-1709, Oct. 1993. [43] J. A. Mezak, and G. D. Vendelin, CAD design of YIG tuned oscillators, Microwave J., pp. 92-98, Dec. 1992. [44] R. K. Mongia, On the accuracy of approximate methods for analyzing cylindrical dielectric resonators, Microwave J., pp. 146-150, Oct. 1991. [45] R. K. Mongia, Resonant frequency of cylindrical dielectric resonator placed in an MIC environment, IEEE Trans. Microwave Theory Tech., Vol. MTI'-38, pp. 802-804, June 1990.
3. Microwave Resonators
93
[46] R. K. Mongia and R. K. Arora, Equivalent circuit parameters of an aperture coupled open resonator cavity, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 1245-1250, Aug. 1993. [47] R. K. Mongia and P. Bhartia, Easy resonant frequency computations for ring resonators, Microwave J., p. 105-108, Nov. 1992. [48] M. J. Niman, Comments on "Whispering galley dielectric resonator modes for W-band devices," IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 1035-1036, May 1992. [49] J.-S. Sun and C.-C. Wei, Iterative method provides accurate resonator analysis, Microwaves & RF, Vol. 30, pp. 70-79, Dec. 1991. [50] F. Tefiku and E. Yamashita, An efficient method for the determination of resonant frequencies of shielded circular disk and ring resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 343-346, Feb. 1993. [51] C. Vedrenne and J. Arnaud, Whispering-gallery modes of dielectric resonators, lEE Proc., Part H, Vol. 129, pp. 183-187, Aug. 1982. [52] A. K. Verma and Z. Rostamy, Resonant frequency of uncovered and covered rectangular microstrip patch using modified Wolff model, IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 109-116, Jan. 1993. [53] W. Xi, W. R. Tinga, W. A. G. Voss, and B. Q. Tian, New results for coaxial re-entrant cavity with partially dielectric filled gap, IEEE Trans. Microwave Theory Tech., Vol. MTT-40, pp. 747-753, Apr. 1992.