Table Of Content~IEEE TRAN SACTI 0 N S ON
MICROWAVE THEORY
AND TECHNIQUES
A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
SEPTEMBER 1995 VOLUME 43 NUMBER 9 IETMAB (ISSN 0018-9480)
PART I OF T\\'0 PARTS
PAPI'RS
TE Modes of an Axially Multiple-Grooved Rectangular Waveguide ................. K. P. Ericksen and A. M. Ferendeci 2001
High Frequency Performance of Multilayer Capacitors ..................................... A. T. Murphy ami F J. Young 2007
Coplanar Waveguides and Microwave Inductors on Silicon Substrates ................................................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . A. C. Reyes, S. M. EI-G/wza/y, S. J. Dom, M. Dydyk, D. K. Schroder, and 1-1. Patterson 2016
Method of Moments Analysis of Anisotropic Artificial Media Composed of Dielectric Wire Objects ................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. E. Peters and E. H. Newman 2023
Contour Integral Method with Fringe Complex Images for the Rapid Solution of Patch Resonators of Arbitrary
Shape ........................................................................ A. A. Omar, Y. L. Chow, and M. G. Stubbs 2028
Direct Extraction of Equivalent Circuit Parameters for I Ieterojunction Bipolar Transistors .............................. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. -1. Wei and J. C. M. Hwang 2035
Scattering at the Junction of a Rectangular Waveguide and a Larger Circular Waveguide ............................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. H. MacPhie and K.-L. Wu 2041
A Full wave CAD Tool for Waveguide Components Using a High Speed Direct Optimizer ............................. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Alessandri, M. Dionigi, and R. Sorrentino 2046
A lligh Accuracy FDTD Algorithm to Solve Microwave Propagation and Scattering Problems on a Coarse Grid
............................................................................................... J. B. Cole 2053
Inverted Stripline Antennas Integrated with Passive and Active Solid-State Devices ........ J. A. Navarro and K. Chang 2059
Mode Conversion and Leaky-Wave Excitation at Open-End Coupled-Microstrip Discontinuities ........................ .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. L. Cina and L. Carin 2066
Development of Self Packaged Iligh Frequency Circuits Using Micromachining Techniques ........................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. F. Drayton and L. P. B. Katehi 2073
Generalized TLM Algorithms with Controlled Stability Margin and Their Equivalence with Finite-Difference Formula-
tions for Modified Grids ..................................................... M. Celuch-Marcysiak and W. K. Gwarek 2081
Examination, Clarification, and Simplification of Modal Decoupling Method for Multiconduetor Transmission Lines
............... ... ... ......... .......... ....... ....... .... ......... .. G.-T. Lei, G.-W. Pan, and B. K. Gilbert 2090
Dispersion Analysis of the Linear Vane-Type Waveguide Using the Generalized Scattering Matrix ..................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. S. Best, R. J. Riegert, and L. C. Goodrich 210 I
Mode-Matching Analysis ofTE -Mode Waveguide Bandpass Filters ............. A. Melloni, M. Politi, and G. G. Genti/i 2109
011
(Continued on hack col'er)
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IEEE TRA 'SACI'IONS ON ~IICROWAVE THEORY AND TECHNIQUES. VOL. ~3. NO.9. SEPTEt-IllER 1995 2001
TE Modes of an Axially
Multiple-Grooved Rectangular Waveguide
Kurt P. Ericksen, Member, IEEE, and Allan M. Ferendeci, Member, IEEE
Abstract-A method is developed to calculate the TE mode electromagnetic fields. The geometry can be thought of as the
fields and cut-on· frequencies of an axially multiple-grooved convolution of the rising-sun type axially grooved cylindrical
rectangular (AGR) waveguide. A low frequency AGR waveguide
waveguides used in magnetrons ]6]. Since the optimum inter
is used as a part of a microwave cavity and the cut-off frequencies
action always occurs with the TE modes of the cavity and the
arc measured in order to verify the analytically derived results.
Excellent agreement bet ween the mcasUJ·ed and calculated values interaction with TM modes is insignificant with regards to the
provide the basis for the design of waveguides for millimctct· wave gyrotron applications, only the calculations and experimental
applications. The method has also been extended to a waveguide verification of the cut-off frequencies of the TE modes for
with multiple grooves cut into two of its broad wall parallel
the AGR waveguide arc presented. The type of waveguide
surfaces.
presented here is completely different than the usual slow
wave periodic structures which contain grooves transverse to
I. INTRODUCTION the direction of propagation.
T HIS PAPER presents a method for finding the TE modes The waveguide shown in Fig. I is made up of two regions.
of an axially grooved rectangular (AGR) waveguide. As Region I consists of the main body of the waveguide and
hown in Fig. I. a waveguide of this type has N uniformly Region II is made up of N uniform axial grooves cut into one
spaced axial grooves cut into one of the broad surfaces. A of the broad surfaces (singly-grooved) of the waveguide. In
particular application for this type of waveguide is a cavity for order to find the cut-off frequencies. solutions to the Helmholtz
the interaction region or a high harmonic rectangular gyrotron equation which satisfy the proper boundary conditions have
[I J. been found in the two regions. Although the analytical results
The output frequency of a gyrotron is directly proportional can be derived in a closed form. only a numerical solution
to the axial magnetic field ]2]. Operation of the gyrotron at m i 1- of the dispersion equation is possible as a result of the com
limetcr wavelengths require superconducting magnets. Since plicated geometry. As expected. solution for the waveguide
gyrotron arc quasi-relativistic devices. harmonic operation has problem occurs only for the discrete values of cut-ofT wave
also high efficiency 13]. Using higher harmonics one can utilize numbers. These arc then used in a cavity configuration to
lower magnetic fields which are within the confines of conven verify experimentally the resulting TE cut-oil frequencies for
tional electromagnets. It has been proposed that by using an the waveguide.
axi encircling pencil beam in an axially grooved magnetron In Section II. the solutions to the Helmholtz equation and
type cylindrical geometry, the high harmonic operation of the derivation of the corresponding fields and the resulting
the gyrotron is highly enhanced ]4]. As the current density dispersion relation arc given. In Section I I L the numerical
increases. the space charge effects in the pencil beam become solution for the cut-off frequencies of various TE modes as
dominant, leading to large velocity spread, thus making the well as the electric field plots arc presented. The solution is
operation at the higher power levels limited. Replacing the extended to the doubly-grooved (grooved on both broad sur
axially grooved cylindrical geometry and the pencil beam by faces) AGR waveguide. Section IV includes an experimental
an axially grooved rectangular waveguide and a rotating ribbon verification of the cut-off frequencies of the AGR waveguide
beam. some of these problems can be reduced ]SJ. The grooves by using the resonant modes of a test cavity made from the
enhance the interaction between the electrons and the fields, given waveguide.
and at the same time reduce the number of competing modes.
The mode excitation is controlled by the dimensions of the II. FIELD EXPRESSION AND THE DISPERSION EQUATION
axial grooves. Use of the ribbon beam also allows the spread In Fig. I. o is the .r-dimcnsion and b is the y-dimcnsion
of the same current over a larger cross sectional area. thus of the uniform section of the waveguide. '2£ is the unit cell
reducing the space charge effects. length. 21\ ' is the groove width. and d is the groove depth.
When used in this manner, a quasi-relativistic sheet electron The last three dimensions are assumed to be identical for
beam injected into the AGR cavity interacts with the cavity each unit cell. There are N grooves (.V unit cells). The
length L is found from L = oj2N. Although for the general
J\lanuscript received August I 0. 1992: rcvi;ed May 25. 1995. Thb work i'
supported in part by Air Force Office of Scientific Research. waveguide problem. there arc no restrictions on the waveguide
The 3uthors arc with the Electrical and Computer Engineering and Com cross sectional dimensions. application to a AGR gyrotron
puter Science Department. University of Cincinnati. Cincinn3ti, OH -15221
imposes restrictions on some or these waveguide dimensions.
USA.
IEEE Log Number 9-113~21. For example. in this case, b should be greater than '2 R 1-. where
00 I 8-9~80/95$0~.00 © 1995 IEEE
2002 IEEE TRANSACTIONS ON ~IICROWAVE TIIEORY AND TECIINIQUES. VOL. ·B. NO. 9. SEPTE~IOER 1995
(6)
tt=-
b (7)
u=-
where
i- 2L~ (8)
Fig. I. Transvcr;.e cross section or the AGR waveguide.
(9)
lh is Ihc Larmor radius of the electrons and a should be To determine the expansion coefficients C, in (5)-(7).
slightly greater than the width of the interacting sheet beam. the boundary condiiion which requires the electric fields of
Solution of the Helmholtz equation \ 21I = + 1.-2 H = = 0 is the two regions to be equal at the groove entrance is used.
=
found for the two regions separately. In region II. fields that Substituting y b into (I) and (5). and multiplying both sides
atisfy the necessary boundary conditions are of the form 151 by cos(k.,.,,:l:) = cos(m1r:rjo) and integrating from 0 to a
results in
;' _ E sin [k,.,(d + lJ -y)] .. [i(2q- l)1r] (I)
1
-'.r - o S•i ll ( k,.0d ) (OS 2JV k2 E (-l)"'N sin (11~11.)
JBJ 1=1 == 0.i kco Eo COS [ k,:0 ( d + b - Y)] COS i(2q- l)7r] ((23)) C'1 1 -_ { jw"cJoL k...yJo1 1 1m sin (ky,b)d if 11 = 2mN ± i
Wfl Sill (k,.,rf) [ 2JV 0 otherwise (I 0)
where k,. is the equivalent rectangular waveguide cut-off where m is an integer.
0
wave number. q is the groove number. and i is an integer Substituting (I 0) into (5)-(7) gives the expressions for the
that deicrmincs the phase difference between Ihc grooves. The fields in the main body of the AGR waveguide. These arc
term i is dcfi ned by
L
E.r =Eo (-1)'11JV
(4)
HI=-
sHuecrcee sis =ive 0 . g·r ·o ·o.v Nes .- F1o ar nid =\li 0i.s \tIfh e= p h0a. steh ed ieffleercetnricce fibeeltdw eheans · { sin(k'1.1.c +1 71.r, .W)(.O S (h. J ..,- .1J )s-Si.-lnli-((/k->y1=11,1+-.-1IyJJ-))) }
the same magnitude and phase at the entrance to every groove. + sin(/.:.,-.11- W) cos ( k.,.ll- .1: )si. n(k,.1 11 (I I)
Also. the value for EJ. at the groove entrance is a maximum. II 7r Sill (/,; IJ)
1111
This results in maximum electron beam interaction when
the waveguide is the interaction region of a high harmonic
rectangular gyrotron. This case is referred to as the 21r mode 111=-
and the AGR cavity is designed so that it supports this mode. · -kJ+.II-+ s.11 1( k.,.,, \1 . )sl.l t (k.,., .., ;) cos(/.> y(11 +.lJ)
The other modes arc designated by their effect on 1]1 with { n 1r ky11- Sill ky11-IJ)
the appropriate values for i and N substituted into (4). For k.,. cos(k,,,-y)
example. with i = 1. N = 3. (4) gives \jt = 1r /3. which is +-n_ 11-1-rS •i ll ( k.,.11- ll1. ) Sl• ll (k.,.,-.r) ky,- sin. (ky - b)
referred to as the 1r /3 mode 171. 11
( 12)
For !J = IJ. (I) gives a constant electric field across the f (
!.:~oEo
groove entrance. This is an approximation to the actual electric II== -1)'" N
field. A more accurate assumption for the fields. which takes }Wfl lll=-
into account Ihe fringing fields at the entrance to the groove. is
sin(kx11+W). (. ·) cos(ky - !J) }
given in the Appendix. However, it is shown in the Appendix . { + cos f..kll" ./. ---'-.- "-,11.----'--..,...
that if the groove depth to groove pitch ratio rl/2 L is small. the n 1r ky11+ Sill (ky11-b)
constant field approximation siill gives very accura1c results. + si11(!.:J_., - W) cos ( /.:.,. :r ) cos(.k y11-.'J) } (13)
ext. it is necessary to find the field expressions in Region 71 7r 11 /,·!JII Sill (k!JII b)
I. For a waveguide in cutoff. the transverse components of the where
magnetic field arc zero. Therefore. the TE field components
in Region I arc E.,.. By, and H=. The field expressions in the
AGR waveguide have Ihe same form as those found for a
rectangular waveguide. The infinite sum formulation for the 11- = 2mJV- i. k.,.,.-
three field components arc 181 (I
( 14)
jWJLf.:!JIIC' ( ) (' )
/,·;, 0 11 cos 1.-.,.11.r sin t>y11.1J (5) The amplitude Eo is determined from the input power supplied
u=- to the waveguide.
ERICKSEN i\ND FERENDECI: TE ~lODES OF <\N i\XIi\LLY MULTIPLE-GROOVED RECTi\ GULAR WAVEGUIDE 2003
The wave number k,. is found by matching the average TABLE I
0
magnetic field in region I to the magnetic field in region II at TEST AGR W,WEGUIDE DI~IENS!ONS
the entrance to each groove 151. The average magnetic field
in region I is found from N 5.7 : I b 0.1~ I I
3 em 2.71 ern em 0.8: ern
1.00
u
""
~
u .00
;":":
0
u
·1.00
FREQUENCY (GHz)
Fig. 2. The dispersion plols for AGR waveguide wilh grooves on one broad
wall for d = 0.8·1 em and d = 1.0 em. (- ) 2rr (i = 0) mode. (- - -
= =
The magnetic field for region II at the groove entrance -)rr/3 (i 1) mode. and (····) 2rr/3 (i 2) mode.
is found by substituting y = b into (3). Equating these
two expressions gives the dispersion equation for the AGR TABLE II
waveguide CuT-OFF FREQUENCIES FOR" S!:-.;GLY-GRoovEo
,\ND DOUIJLY-GROOVED WAVEGUIDE
(n+1rw)
Loo (_ S.l ll-? --- single double
=
W 1) '" (n+:wa )?- cot k(kuy,'11- b) mode i k Eeo (GHz) Eeo IGHz)
2L , =- 0 I 5.1666 4.8941
2 9.2994 10.526
oo S.l ll?- ( -n--7rW-) 3 11.999
W L ( a
+ 2L - l) rt/3 I 2.5327 5.3786
II/ ?
m=-= c~-:wr 2 5.6927 10.841
3 9.5519 13.611
cot (ky,-b) 2rt/3 2 I 5.0165 6.5684
( 16)
2 7.0163
3 11.443
Ill. CUT-OFF FREQUE 'CIES 4 12.516
In order to complete the derivation of the AGR waveguide
fields, the waveguide cutoff wave numbers. k,. are to be Table II gives the calculated cut-off frequencies found from
0.
determined. This is done by solving ( 16). The simplest way is the numerical solution of the dispersion ( 16) for the ingly
10 plot both the right and left hand sides of ( 16) on the same grooved AGR waveguide for N = 3.
graph as a function of frequency. The intersection points of A second plot of - cot (k .. d)/k .. for a different groove
the two curves give the cut-off frequencies for that mode. The height d = l.O em is also shown in Fig. 2. Changing the groove
approximate result for the frequency from the plot is used as a height changes the cut-off frequencies of the waveguide when
starting condition for a more accurate numerical solution using all the other physical dimensions are kept constant. If pro
the bisection method [91. The corresponding modes are labeled visions are made so that simultaneously movable mechanical
as TE;k· modes where k stands for the kth cut-off frequency sliding shorts can be introduced into the grooves, the cut-off
associated with the ith mode. frequency of a given mode can be changed by varying the
The AGR waveguide dimensions used when ploning ( 16) depth d of the grooves. This allows the resulting cavity for
are given in Table I. The dispersion equation plots are shown the rectangular gyrotron to be frequency tunable.
in Fig. 2 for i= O(IJ! = 27r).i = l(IJ! = 7r/3). and i = 2(1Ji = Once 1.-,. for a given mode is found. the corresponding
0
21r /3) modes up to 15 GHz. The cut-off frequencies !co arc transverse electric fields for the waveguide arc known. They
found from kco by are then used to graphically plot the transverse electric field
lines for each mode II OJ. The AGR waveguide is periodic in
( 17) the transverse (.1:) direction with a unit cell of length 2£ for
the 21r mode. Therefore, the field distribution is symmetric
where c is the speed of light. over each of the unit cells for the 21r mode. Electric field
IEEE TRANSACTIONS ON ~IICROWAVE TIIEORY AND TECHNIQUES. VOL. -13. NO. 9. SEPTE~IBER 1995
= =
plots corresponding to the k 1 and !.: :J modes of the
271' mode for a singly-grooved waveguide are shown in Fig.
3. It is apparent that the fringing fields in the vicinity of the ___
grooves arc high for the i = 0.!.: = :J mode. Thus. for a high :--
harmonic gyrotron operation, this mode is therefore preferred
= = =
over the i 0.!.: I and k 2 modes.
To verify that the presence of the grooves. in addition 'E 2~------1
~
to increasing the interaction efficiency, reduces the number >
of competing modes associated with the TE modes of a
standard oversized waveguide, a similar derivation is applied
to a rectangular waveguide with identical axial grooves at
the two wider walls. Assuming that the fields at the grooves
satisfy the same boundary conditions as the singly-grooved oo... ......- ~. .....,...- ~. .......... J...J
waveguide results in the following expressions for the fields X (em) X( em)
and dispersion equation. The field expressions are (a) (b)
L00 Fig. 3. Transverse electric field lines for the singly-grooved i\GR waveguide
EJ. =Eo (-l)"'JV :2..- modes: a) i = 0. k = I mode, f,., = .j, JGGG Gl-lz, and b) i = 0. k = 3
mode. f,. . , = 11.990 GI-lt..
til=-
!iill (/.;J'/1+\V) • ·(/. ·) CO!i(ky +,1J)
. { u+'il' cos "'k,+·'· cos(ky 1•1bI 2)
11
+ siu (/n.-.-•. 111 1' W) cos (,', J.,-:z:) cco·so (s k(kyy,-,-bI: tJ 2) ) } <1 8) 1.00
i!!u
~
Ill=- g""u I
,I
,I
(.) -1.00
I
I
I
I
I
I
-2. 0 0 '=-=---=-'='.!...,_-:'+.c-..___-::":-:----..,:-::-'::-::----..,="""':-::----1': 7
.00 12.50 15.00
Fig . .J. The dispersion plots for AGR waveguide with groove' on both broad
walls ford= 0.8·1 em.(-) :2;; (i = 0) mode.(----) ..-f3 (i =I) mode.
and (-· · ·) :2..-/3 ( i = 2) mode. TE,." modes of a groove less rectangular
waveguide are also ploucd below !he frequency scale.
for the singly-grooved waveguide. In Fig. -k corresponding
TE""' cut-ofT frequencies for a groovclcss waveguide with the
same a and b dimensions are also shown below the frequency
axis to compare the cut-ofT frequencies of the waveguide with
and without grooves.
The calculated cut-ofT frequencies for a doubly-grooved
The dispersion equation is AGR waveguide arc al o listed in Table II. The corresponding
electric field distribution for the k = 1 and k = :J modes
!'Ot (kcod) N of the 271' mode for a doubly-grooved waveguide with all the
k,.o \111!' same dimensions given in Table I, except with b = 5.15 em
nr=-
is plotted in Fig. 5.
IV. EXPERIMENTAL VERIFICATION OF CUT-Of-F MODES
N
+ - """" ( - 1)"'+ I The theoretically calculated cut-off frequencies for the AGR
11171' ~
waveguide arc verified experimentally using a test cavity.
Ill=-
=
siu2 (k,.,- IV) tan (ky,-bl2) A short length of the AGR waveguide (JJ 6.03 em) is
(21) machined with the transverse dimensions given in Table I.
Two shorting plates arc placed at each encl. A small probe is
Plot of (21) is given in Fig. 4. The dimensions and the inserted into the cavity through one of the side walls to excite
number of grooves on both walls arc the same as those given the cavity as shown in Fig. 6.
ERICKSEN AND FERENDECI: TE MODES OF AN AXIALLY MULTIPLE-GROOVED RECTANGULAR WAVEGUIDE 2005
4 4 TABLE Ill
CALCULATED AND MEASURED TE,k·
D Jl
RESONANT FREQUENCIES I·OR TilE TEST CAVITY
3 3
v "---
~ theoretical measured
2 2 <-::::> ~ i k 1 lfcJ ik lfcJ ikl lfc) ikl
27t 0 I 9.2994 9.6264 9.6225
3 11.924 11.985
'E 'E lt/3 2.5327 3.5502 3.5525
0 0
>~ ~ ~> 3 7.8817 7.87
-1 -1 5 12.695 12.678
,.._<::::> __ 3 I 9.5519 9.8706 9.8875
-2 -2 =- 2 10.770 10.84
t-u'-.~ lf 3 12.122 12.13
-3 -3
resonant frequencies or the cavity which are close to the
-4 -4
0 1 2 0 1 2 predicted resonant frequencies arc also included in Table Ill.
X (em) X (em)
(a) (b) V. DISCUSSION AND CONCLUSION
Fig. 5. Transverse eleclric field lines for lhe doubly-grooved AGR wave
= = = = A comparison of the calculated resonant frequencies to
guide 2;;- modes (1.1 5.15 em): a) i 0. ~- 1 mode. f,.o 2.6233 GHz,
and b) i = 0. k = 3 mode, feu = 13.633 Gl-lz. the mea ured frequencies given in Table Ill shows excellent
agreement. The greatest percent difference between calculated
and measured results was 0.9%. The average percent difference
for the test cavity was only 0.2%. Considering the constant
field approximation at the groove opening. the measured
frequencies are in close agreement.
Both of the dispersion relations given by (21) and (27)
reduce to the cut-off wave numbers for a groovclcss rectan
gular waveguide as d and W approach 0. As d approaches
0, the left side of both equations approach . These arc
satisfied if both kuu+ and k1111_ arc equal to llm/b on the
right side, This leads to the usual waveguide cut-off wave
numbers k?, = (n1rja)2 + (m7r/bf .
0
Fig. 6. AGR cavi1y for measuring rcsonam frequencies. When the number of grooves is equal to one. axially grooved
waveguide presented here is equivalent to a channel waveguide
J,.
lll j. For a channel waveguide, the cut-off frequency is
As a result of the shorting plates placed at the two ends of
the waveguide. an axial standing wave is generated between calculated and compared with the cut-ofT frequency f,.Jr~; 1o of
the TE mode of a groovclcss rectangular waveguide with
these two planes. The resulting cavity resonance frequencies 10
the same a and b dimensions. For the following parameters
are given by
b/a = 0.-155. (d + b)/b = 0.5. and ll"/L = 0.2:..1 with
?]1/2
Jc,•kl = 27r ~1 [( kco)?i k + (Dl7r ) - (22) cILh a=nn 1e.l9 w1 aevmeg, utihdee r1a1ti1o1 . fUcs],in, g10 t/ hfec siasm eeq upaalr atmo e1te.0rs9. f(o 1r6 )t hies
plotted for N = 1 and the resulting lowest order mode cut
Here l is an integer and D is the lenglh of the cavity in off frequency fc of the AGR is calculated. For this ca e. the
the z direction. The corresponding modes can be identified as ratio of fciTEo/ fc is found to be 1.07. Thus. the calculations
1
TE;kl resonant modes for the cavity. Here (kco),. corresponds presented here also agree with the results of the channel
to the cut-off wave number of the AGR waveguide for the kth waveguide.
zero of the ith mode. As can be seen from Table II, if the same number of
The TEikl cavity resonant frequencies are calculated from axial grooves are introduced on the second broad-surface
(22) up to 15 GHz using the previously calculated cut-off of the waveguide. the number of TE modes supported by
wave number kco values given in Table II. These are tabulated the waveguide decreases. thus further reducing the mode
in Table Ill. An HP-851 0 microwave network analyzer is competition problems associated with oversized waveguides.
wept in frequency and the resonant frequencies of the test Extending the calculation presented here to millimeter
cavity are recorded. These frequencies are compared with the wavelengths is straight forward. For example. if' operation at
theoretically calculated results. Although there were additional 90 GHz and sixth harmonic is desired, the Larmor radius for
resonance attributed to other modes such as TM modes, no the fundamental cyclotron frequency of 15 GHz is calculated
attempt was made to identify these resonances. The measured for a given beam energy. The dimension b is chosen slightly
2006 IEEE TRA SACTIONS ON ~IICROIVAVF. TIIF.ORY AND TECIINIQUES. VOL. ~J. NO. 9. SEJYrE~IOER 1995
larger that 21h. Once the width of the rotating sheet beam
is given. the overall dimension a and the number of grooves
N can be chosen. By using ( 16) and by a repeated numerical
calculation process, the groove width I II and the groove
depth d can then be determined to provide the desired cut-ofT
frequency of the TEak mode at 90 GHz for the 21r mode 111=-
=
(i 0). Since various combinations of IV and rl will satisfy
this condition. it is very important that the cut-off frequencies
of the other TEoj modes (j = 1 · · · k - I) of the grooved
where .1 is the zeroth order Bessel Function of the fir t kind.
waveguide do not coincide with any of the lower harmonics 0
The difTerence in the resonant frequencies found from ( 16)
of the fundamental cyclotron frequency to prevent oscillations
and (25) is within 5% if the ratio d/2£ < 0.15. The ratio dj2L
at lower frequencies.
is groove width to groove pitch ratio.
The close agreement between the experimentally measured
cut-off frequencies with the theoretically predicted values as
REFERENCES
well as the behavior of the field lines at the boundaries of the
waveguide provide a proof to the validity of the assumptions I I I A.M. Fcrendcei ... Rcclangular cavi1y high harmonic gyro1ron amplifier:·
made with regards to these calculations. in IEEE fv/7T-S Dig .. Boswn. MA. vol. N-9. May 1983. pp. 430-131.
121 L. J. Craig ...S ialus of gyrolron dcvclopmen1:· J. Fusion Energy. vol.
6. pp. 351-360. 1987.
APPENDIX 131 K. R. Chu ...T heory of elc.:e1ron cyclolron maser inleraclion in a cavily a1
I he harmonic-frequencies:· Phys. Fluids. vol. 2 I. pp. 235-l-23M. 1978.
A common practice for finding the solutions at a waveguide HI Y. Y. Lau and L. R. Barnell. ··Theory of low magne1ic field gyro1ron
discontinuity i. to assume a solution at the discontinuity. In (Gyromagnclron):· Int. J. Infrared and Millimeter \\1n·es. vol. 3, p. 6 I 9.
1982.
this paper. a constant field at the entrance of the grooves is 151 C. C. Han and A.M. Ferendeci ... Non-linear analysis of a high harmonic
assumed. However. a more accurate assumed solution is the reclangular gyr01ron:· Int. J. Electron .. vol. 57. pp. 1055-1063, 1984.
161 N. M. Kroll and \V. E. Lamb ...T he resonanl modes of lhe rising sun
qua i-static gap field also referred to as the electrostatic fringe
and o1her uns1rapped mag111.:1ron anode blocks ... J. tlppl. Phys. . vol. 19.
field 141. 161. 171. no. 2. pp. 166-186. 19-18.
Assuming this type of field at the entrance to the groove 171 R. G. E. Huller, /Jew11 and \\'m·e Electronics in Micrm.-m·e 7itbes. lsi
ed. Princ..:lon, NJ: D. Van Noslrand, 1960.
changes the constant term Eo of (I) and (3). There is now an
181 1-l. Mot1.. Electromagnetic Problems of 1\/icrmo·m·e 111eory. lsi eel..
.r-dependence to the fields. This is given by 161 Melhuen & Co. . Lid .. 195 I.
p 191 S. Chapra and R. P. Camale. Introduction to Computing for Engineer.~.
New York: McGraw-Hill. 1986.
- (:t:- (23) I 101 P. E. Moller and R. H. Macphi..: ...O n lhc graphical represenlation of
2
(2q- 1)£) elcclric fields in waveguide:· IEEE Trans. Micro11·m·e Theon· Tech. . vol.
w
J 33. pp. I 87-I 92. 1985.
I I II R. J. Viltnur and K. Ishii ...T he channel waveguide:· IRE 7i·ans.
Micrmmn' Themy Tech., vol. I 0. pp. 220-22 I. I 962.
where P is a constant. The fields in region II are now of the
form
P sin [kco(d + b-y)]
E_,.= _( ·r _(
Kurt P. Ericksen (S'86-M'93) n:ccived lhe B.S.
2q _ 1)£)2 ~ill (k,.0d) and Ph.D. degrees frorn1hc Universily of Cincinnali
1 IV and M.S. d.:gr.:c from Ohio S1a1e Universily in
I 985. I 988. and 1992. rcsp.:c1ively. His graduate
i(2q- 1)7r] research was in 1he areas or semiconduc10r physics
·co~ [ 2N (24) and applied clec1romagne1ics. Panicular focus was
on high power millimc1cr wave lubes.
(25) In 1993. he joined Molorola Paging Producls
Group, Schaumburg. Illinois, where he has worked
H.= on 1ram.mi11er design for paging infraslruciUrc ap
pl ical ions.
i(2q- l)7r]
. ('0~ [ 2N . (26) Allan l\1. Fcrcndcci (M'63) is a Facul1y Member
a1 1he Elcclrical and Compul.:r Engineering and
Equation (26) is used to find the new expansion coefficient Compu1er Science Deparlmenl al lhc Univcrsily of
Cincinnali. 1-l.: is also 1hc direc10r of 1he MillimeiCr
C" for (5)-(7). The new resonant condition is found using the
Wave Elcclronics Labora10ry. His imcresls arc in
same procedure given in Sections Ill and IV. The resulting microwaves and millimeler wave devices and cir
resonance condition for the fringing field assumption is cuils. microwave applications of high lcmperalllrc
supcrconduc10rs. clcc1ro-op1ics. elcc1ron beam de
vices and gyro1rons. In addilion 10 being aulhor
~Ill ( 1/ +(7LI \1!1 ) and co-au1hor or various p:~p.:rs and conference
"' ( - 1) ---'--:----~ proceedings. he is also lhe aulhor of a h:xt book
~ Ill n+7r Phy.~ical Foundmions of Solid State and Eil•ctmn De1·ices and c:o-cdilor of I he
llt=-cx:> book Atomic and Molecular Processes in Controlled Thermonuclear Research.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. ~3. NO.9. SEIYrE~IBER 1995 2007
High Frequency Performance
of Multilayer Capacitors
Arthur T. Murphy, Fe!loiV, 1£££, and Frederick J. Young
Abstract-The high frequency performance of capacitors is A AIR B
related to their geometry and material properties. By considering CURRENT IN TOP ELECTRODE
gcactmtoyaoop uppompewldalt pic ieoeluiladaflttg -oy ecdrreorteos-ron,ea c nmuicadseamntceepudntdeadi tn oct yhtnimeb t eodetur or teilwnsptsc ifooae-l(lunpceioveonaaignncn Lycietd.Ct e xuoT )fpcro rhsetefao er qsgird u mm redeotseenricutnsachrnttniiroadbeisld sbem ,psdu r il eetseaesssiqnndiuteuoa lsntibtes,vh l liateeslealci hesnntelntred iditt f c ieaaxaarcnrtlnai ucratascur llayeuyyrpssesist,p ete seslsaa mi nneoisidddssf, //// //// /// /// //// //// /// /// /// / /// / / / D// / I// /E / '/ /L / E/ / C// / //T / '/ R / / /I C// / / // / / / / / / / // / / // / // / / / / // / / // / // / / / / // / /
obtained. CURRENT OUT BOTTOM ELECTRODE
c AIR D
I. I TRODUCTION TYPE A CONNECTION
(a)
A T VERY low frequencies. capacitors exhibit ideal be
havior and have only three properties. These are the A AIR B
capacitance. dissipation factor. and voltage rating. Under such CURRENT IN,_ TOP ELECTRODE
ccacinoonidrdnncu ndueceumtvcciesvtteoetrady rn secc tfeoiufses r crtuehitnt ne ia tdm ncopdapac posteh artv cnaeiionntnotst .tri ud mAapealls lauyt ht eteherbse e chacafoornpewmdaq ecuttihhseto eenar c epyhtx rraeartsesin sersanenmnsac , lei isen csacoiicorrfechn au o siltlethii anneidgesr ' //// //// /// /// //// //// /// /// /// / /// / // D// / I// /E / / //L / E// C// / / //T / /R/ / /I C// / // / / / / / / / / // / // / / / / / / / / // / / // / / / / / / /// //
of sorts. Then capacitors show resonances and antiresonanccs CURRENT OUT
BOTTOM ELECTRODE
which may be innuenccd by the lead placement or the presence c AIR D
of ground planes. Although high frequency phenomena in ca
TYPE B CONNECTION
pacitors have been described before 14], 15], and llll they have (b)
not been analyzed using the recent advances in the calculation
Fig. I. Capaci10r conn.:ction topology.
of electromagnetic fields. These new methods allow an exact
treatment of distributed multilayer Structures I 121. 1131. It is
geometry and material properties to enable an analysis based
our goal to examine some high frequency phenomena from
a fundamental view point utilizing the distributed analyses upon physical principles. Our new methods arc applied to
capacitors measured by Boser and Newsome 121 and Burn
of ICO SIM II j (computer aided interconnection simulation)
and appropriate physical models of capacitors. In all cases the and Porter 131w hich have 2 and 6 plates. respectively.
parameters of the phy ical models of the capacitors arc derived
from the capacitor geometry and the physical properties of the B. f111portant Ne11· Conclusions
materials used. I) The Influence of Connection Topology: In a two elec
trode capacitor the leads may be auachcd at any point on the
plate as long as each plate has but one lead connected to it. At
II. SUMMARY OF THE RESULTS OF NEW METHODS
de and low frequencies all . uch connections made at arbitrary
locations on the plates yield the same value of capacitance. At
A. General Purpose
higher frequencies, when electrode inductance and resistance
It is our goal to verify our new methods of capacitor
become important, the exact nature of the connections must
analyses with experimental results given in the literature. be considered. In this work we consider the two simplest
Although many papers present experimental results for various
connections. In a type A connection current enters and leaves
type of capacitors. very few give sufficient detai I about the on the same side of the capacitor, while in a type B connection
current enters and leaves on opposite sides. Fig. I shows both
Manuscripl received October 1-1. 1992; revised May 25. 1995.
A. T. Murphy is with Central Research and Development. Experimelllal type A and type B connections.
Station, E. I. du Pont de Nemours & Co. Inc., Wilmington. DE 19880-017-1 We show that type A topology leads to a high resonant
USA.
frequency and high equivalent series resistance (ESR). The
F. J. Young is at 800 Minard Run Road. Bradford, PA 16701-37t8 USA.
IEEE Log Number 9-113-116. type B connection yields a low resonant frequency and a low
0018-9480/95$0-1.00 © 1995 IEEE
2008 IEEE TRANSACTIONS ON ~IICROWi\VE TIIEORY AND TECHNIQUES. VOL. .IJ. NO.9. SI:PTE~IBER 1995
8.0
I
7.0 \
6.0
a:<n \
oz I TYPE B MAGNITUDE OF IMPEDANCE
-V:lSO 5.0 :1
:O<t
:oz:a-: 4.0 \ti
-w I
zWU<z-!l' 3.0 II f \_TYPE A MAGNITUDE OF IMPEDANCE
<t<t 2.0 ···"'-- _.... v
Ow I I
:aWO.<(:z/t): 1.0 II fl ·" I .-.-· ·r·
-a. 0.0 ~: TPHYAPSEEB ANGLE . ..- ..- SPILLAVTEERS P WAILTLHA ADBIUOMU T K = 60 (BaNdTi03
-1.0 ..::.+-·-· 0.1 mil THICKNESS TAM-COG 500H)
TYPE A PHASE ANGLE
-2.0 ' ' f'ig. 3. Bos~r and N~w~omc·~ 121 two plate capacitor geom~try with all
0.00 1.00 2.00 3.00 4.00 5.00 6.00 dimcm,ion~ in mib.
FREQUENCY IN GIGAHERTZ
f'ig. 2. Calcula1cd impedance and pha~e anglc-4:omparbon of 1ypc A and
type 13 connection\ for tlu: Bo-.er and New~ome 121 NP02S I capacitor. ········MEASURED
--CALCULATED
103~
ESR. This is illustrated in Fig. 2 which shows our calculated
~
impedance versus frequency curves or one or Boser and u E
z
cwsomc·s 121 capacitors for type A and B topologies. In <t
0
the type A connection the resistive losses cause the minimum aw.
in the magnitude of the impedance to occur at 3.05 gigahertz ~
~ 101=-~----
in.tcad of 3.45 gigahertz where the phase angle is zero and
w
the impedance is a pure resistance of 620 milliohms. In a 0
t~oo
type B connection the minimum magnitude or the impedance ~ 100§.,. -___
occurs at 0.639 gigahertz whilst the phase angle is zero at <t
:;:
0.640 gigahertz where the impedance is a pure resistance of 1 o-•'r- --~~_.....~ -----.1
520 milliohms. Note the great difference between resonant 1.0 10 100 1000
FREQUENCY IN MEGAHERTZ
frequencies indicating that a type A connected capacitor acts
as a capacitor over a much greater frequency range than docs f'ig. -l. Comparison of mea\urcd and calculated impedance of the NP02S I
capacitor of Boser and New-,omc (2(.
a type B connected capacitor. The resistance for a type A
topology is about 16lfl. larger than for type B in this case.
frequency is 676 megahertz. In Fig. 4 is a comparison of our
2) The Influence of Ground Planes and Test Fixtures: Each
calculated and the measured impedance. The details of these
electrode of a capacitor has a self inductance that is reduced
calculations arc given in Section III-A. Boser and cwsomc
progressively as ground planes or ground conductors arc
121 also measured an "identical'· capacitor which they called
moved closer. The mutual coupling is not changed as greatly
NP02S2. It had the same resonant frequency but only 310
and capacitance is not influenced. The resistances arc also
milliohms of resistance.
influenced by ground conductors. It is well known in the
measurement of type B connected capacitors that the test
C. Six-Plate Multilayer Capacitor
fixture exerts an influence on the measured resonant frequency.
This is because part of the test fixture is a pipe-like ground Capacitors arc often constructed with several layers of
which causes the capacitor electrode inductance to vary as a dielectric separated by thin conducting layers which serve as
function of its internal diameter. the capacitor plates. In Fig. 5 is shown a multilayer capacitor
We have obtained additional information 161 pertaining to (M LC) constructed and tested by Burn and Porter 131. For
the capacitors of Boser and ewsomc 121 of Philips Labo simplicity. we show the capacitor electrodes or plates to be
ratories at Briarcliff. NY. The two plate P02S I capacitor of zero thickness.
depicted in Fig. 3 is analysed. The area of the top of each dielectric layer is the overlap area
In this capacitor. utilizing a type B connection. current and sets the value of capacitance. Using a value of 18.5 for the
enters at side A of the top conductor and leaves at side B relative permittivity of the layers yields a total de capacitance
of the bottom conductor. Boser and Newsome 121 measured of I 00 picofarad. The influence of the test fixture in which
a resistance or 520 milliohms at the first resonant frequency the MLC is mounted must be considered. The presence of
of 660.6 megahertz and a capacitance of 50 picofarads. Our a ground structure can greatly alter the inductance of the
mathematical models based upon the physical properties of capacitor plates. The closer the ground structure. the lower said
the materials and the geometry yield 525 milliohms resistance inductance is. In Fig. 6 is shown the ILC in the I centimeter
at a resonant frequency of 639 megahertz. Our calculations 10 test fixture. Measurements indicate Burn and Portcr·s 131
in agreement with the original capacitor design indicate that MLC Test Fixture introduces about 0.5 nanohcnry of lead
dielectric losses arc not important at this frequency. If the inductance which is included in our calculation or impedance
conductor. arc assumed to be losslcss the calculated resonant as a function of frequency. We calculate a first resonant
