IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-33, NO. 3, MARCH 1985
which,
by definition
[8], is
cal. For finite
Zn(u)=(
For small
arguments
(V<
–l)-n.ln(jv).
(16)
calculated
271
values of surface resistance,
both
the resistive
the equations
and reactive
portions
impedance.
1) [9]
I_n(v)=
I,,(u) =+(;)n.
IV.
(17)
New
equations,
useful
CONCLUSION
for
the
accurate
Combining (14b), (16), and (17), using only the n = – 1, n = O, complex input impedance of lossy, radial-line
and n = + 1 terms of the summation in (14b), and using the presented. This should lead to an improvement
equality
the predicted
performance
of circuits which
J_n(zl)=(-l)nJn(u)
(18)
=Jm(U)+
$[J~_l(U)–
Jm+l(U)]
(19a)
For
completeness,
notation
of this
the
paper,
is derived
from
(19a) by utilizing
the
in the accuracy
contain
of
these ele-
r,
the recurrence
due
to
Vinding,
using
the
below.
(Al)
sin[+(~rt)–+(ko)l
[10]
.lm_l(u)-Jm+l(u)
(20)
=’2J;(r.J).
1207
~
Zo(kr,)=
Nm(u+ju)
two additional
=Nm(u)+juN;
recurrence
J~(u)
(u).
Jj(kr, )+ Nj(k~)
Jf(kr,)+Nf(krl)
[
JL ( u) is the derivative of Jm( u) with respect to u. Similarly
O(kr,)
= t~-l[NO(kr,)/Jo(
~(kr,)
= tan-l
$(kro)
=
I
1’2
(A2)
kr,)]
(A3)
(21)
[-
(A4)
J1(kr,)/Nl(kr,)]
relationships
(22a)
= –Jl(u)
(A5)
tan-l[- J,(kro)/N,(kro)].
and’
RHFERSNCES
J;(U)
(22b)
=JO(U)–+J1(U)
[1]
N. Marcuvitz,
[2]
Montgomery, R. H. Dicke, and E. M. Purcell, E&,
(Radiation Laboratory Series, Vol. 8). New York, NY: McGraw-Hill,
1947, pp. 240-282.
S. Rarno and J. R. Whinnery, Fteldr and Waves in Modern Radto, Second
Circui~s,
the resulting equations with the substitutions u = ~r and v = – w
are
Jo(h)
= JO(/3r)
.lI(kr)
=J1(/3r)-
(23a)
+ jcwJ1(flr)
.ll(~r)
jar
Jo(&)-~
1
.
[
For the Neumann
[3]
[4]
(23b)
[5]
functions
NO( kr)
= NO(~r)
Nl(kr)
=N1(~r)–
(23c)
+ jarN1(~r)
N1(/3r)
jar
No(~r)–--jjy-
[
1
.
(23d)
The final analysis equations for radial-line stubs with attenuation
result from substituting (23a) through (23d), with the appropriate
arguments, into (5) and the latter, in turn, into (4).
[6]
[7]
Mathematical
w=(~+rO)sin
o
~ .
()
(24)
line$”
in Principles
of
Microwave
Functions
with
Formulas,
Graphs
and
with
Formulas,
Graphs
and
Formulas,
Graphs
and
pp. 375, eq. 9.6.3.
of Mathematical
Tables,
Hundbook
Functions
pp. 375, eq. 9.6.6 and 9.6.7.
of Mathematical
Functions
with
pp. 361, eq. 9.1.27.
S. L. March, “ Wcrostrip
packaging: Watch the last step,” Microwaves,
vol. 20, pp. 83–94, Dec. 1981.
=;attcal
For stripline, the dimension h in (4), (6), (12), and (13) should
be replaced by the ground-plane spacing b.
For rnicrostrip, it is recommended that the relative dielectric
constant t, should be replaced by an effective dielectric constant
ce~~ calculated [11] for a microstnp of constant width w, where
of Mathematical
Tables,
Handbook
[9]
[11]
transmission
G.
Handbook
[8]
[10]
I@CONiMENDATIONS
FORSTRIPLINEAND MICROSTRIP
C.
“Radial
Ed. New York, NY: Wiley, 1953, pp. 395-400.
H. A. Atwater,
“ Microstrip
reactive circuit elements,” IEEE
Trans.
Microwaue
Theory
Tech., vol. MTT-31, pp. 488-491, June 1983.
B. A. Syrett, “A broad-band
element for microstrip
bias or tuning
circuits,”
IEEE
Trans.
Microwave
Theory
Tech.,
vol. MTT-28,
pp.
925-927, Aug. 1980.
‘f Broadband
microstrip
mixer design: The Butterfly
mixer;’
A~c~tion
Note Number 976, Hewlett-Packard
Co., Palo Alto, CA,
hdy
1980.
J. P. Vinding,
“Radial line stubs as elements in strip line circuits,”
NEREM
Rec., pp. 108-109, Nov. 1967.
M. Abrarnowitz
and I. E. Stegun, Eds., Handbook
of Mathemaucal
(Applied
Functions
with
Formulas,
Graphs
and Mathematical
Tables,
Mathematics
Series, Vol. 55). Washington, DC: U.S. Department
of
Commerce, National Bureau of Standards, 1964, pp. 363, eq. 9.1.75.
=hutical
III.
have been
cos[e(b,)-+(bo)]
Zo(kri)lr
Z,n = j
(19b)
equations
are included
(19b)
=Jm(u)+jul;(u).
Using
of
stubs,
APPENDIX
J~(U+jU)
relation
calculation
ments.
results in
Equation
correctly
of the input
Tables,
Plot of Modal
Field Distribution
Circular
in Rectangular
and
Waveguides
C. S. LEE, S. W. LEE, AND S. L. CHUANG
The earliest plots of modal field distribution
in rectanguhu-/cir-
A computer program has been written to test these equations
and to compare the results with the lossless formulation
of
Vinding. For perfect conductors (R. = O), the results were identi.
cular waveguides
were given by Southworth
(1936) [1], Barrow
(1936) [2], Schelkunoff
(1937) [3], and Chu and Barrow (1937) [4].
1This approximation, while providing reasonably accurate results, is not the
formulation
for effective dielectric constarrt used in Super-Compact.
Manuscript
received August 9, 1984; revised October 30, 1984. This work
was supported in part by NASA under Contract NAG3-475.
The authors zre with the Electromagnetic Laboratory, Urriversity of Illinois,
Urbana, IL 61801.
0018-9480/85/0300’0271
$01.oo @1985 IEEE
272
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-33, NO. 3, MARCH 1985
,
,
0
TE,O
,
I
I
2
I
TEO,
TEZO
TE,,
TE2,
TM,,
TM2,
$
I
2
1
3
TE3,
TM3,
TE30
I
TEq,
TE22
TM4,
TM22
3
TE40 TE12
TEOZ TM,z
I
4
TE,0
TE 32
I
TM5,
,
5
TE60TE13
TE03TM,3
1
1
r
I
1
5
TE4Z
TE51
TM32
1
4
TM*2
,
I
TE.,
TE;3
TE
TM61
52
TM52
‘“23
I
6
TE,3
TE70
TM33
I
6
Fig.
1
7
Normalized
modal cutoff frequencies for a 2:1 rectangular
waveguid~.
., ~
-.,
TE50
TE32
TE5, “
TM32
TM51
TM4Z
T-.
TE60
e
TE,3
TE52
E—
—
..
-—_
.,
Z@,
,,
=
---
‘s3’:
❑ ---
--
-I L,.
I to,
TE20
TE$,
TM,,
TE2,
TM21
TE30
TE3,
TM13
TE6,
TM3,
T E.O
‘“61
TM23
TE ,2
TM,,
TE33
..
TE22
TM33
H –-–-–
Fig. 2.
m
.-.
Transverse
modal field distribution
TE.,
TM.,
E—
for a 2:1 rectanguliw
TE02
—
waveguide
TM,,-.
H --––-
(first 36 modes).
.- —-—- ... . . .. ... . .. .... . . .....,.Itkt
lKAN>AL1lUNb
UN
“..,r.
MILKUWAVE
n”.,
.
ltIEUKI
273
TECHNIQUES, VOL. MTT-33, NO. 3, MARCH 1985
.,-
AMJ
0
TE,,
T,M,,
&
L
E
I
2
TE21
TE,2
TM2,
TM,2
TErO
TE02
a
I
TE22
TM22
I
2
TE30
TE03
3
TE3,
TE,3
TM3,
TE3Z
TE23
TM32
TM13
TM23
TE
TE;:
TE40 TM41 TE33
TE04 TM14 TM35
1,
TM,,
TE ~.
TE03
Fig. 4.
text and reference
six or seven modes.
higher
5
modal cutoff frequencies for a square wavegtide.
TEL,
TE22
E—
for
TM24
TM31
TE13
TE02
TE32
TM22
TE4,
TE,2
.. _
plots
I
TE304
TE05
TE43TM43
TE34TM34
I
I
TM12
present
Normalized
to,
TE,O
TE2,
Standard
I
TE42
TE24
TM02
I
4
Fig. 3.
I
3
Transverse
books present
E—
modal field distribution
plots only for the first
applications,
we are interested
in
modes. The purpose of this note is to
complete
TE33
TE31
~——---
In many
order
three relatively
TE,.
sets of plots, namely,
a) plots for
for a square waveguide (first
30 modes)
the first
36 modes in a 2:1 rectangular
b) plots
for the first
4), c) plots
and 6).
TM,,
~-----
waveguide
(Figs. 1 and 2),
30 modes in a square waveguide
for the first
30 modes in a circular
(Figs. 3 and
waveguide
(Figs. 5
274
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-33 , NO. 3, MARCH 1985
c1
a
TE,,
TMO,
I
I
TM.,
TE32
TM22
TE,3
TE71
T
2
~E6,
TM,,
TEO,
TE6,
TE2,
TE3,
TM2,
I
2
TM02
[
3
TE2Z
/
3
TE02
TM12
I
I
4
TE,3
TE6,
51
TE32
TWI
r
4
TM22
,
TE.2
I
TE8,
TM32
TE52 TE91
I
5
Fig. 5.
!
5
,
I
Normalized
r
1
6
modal cutoff frequencies for a circular waveguide,
~..-.
/---.’
,., /, -
-
,-
~(11
. ‘,
\’\”,
!;:
E—-
. . -’ >,, ,,’
~\’\’ - -,
-------‘
@ ----TE,,
TMO,
~“”
Fig. 6.
TE2,
The density
field
@
(Continued)
of the field lines is approximately
strength.
These plots
proportional
are done by a Cyber
to the
175 computer.
@
TM,,
TEO,
TE31
I&5FERENCES
[1]
[2]
[3]
TM2,
TE4,
TE,2
[4]
G. C. Southworth,
“ HWer-frequency
waveguides— General considerations and experimental results,” Bell Syst. Tech. J., vol. 15, pp. 284-309,
ALrr. 1936.
WA.L. Barrow, ‘<Transmission of electromagnetic waves in hollow tubes of
metals,” Proc. IRE, vol. 24, pp. 1298–1328, Oct. 1936,
S. A, Schelkunoff, “Transmission
theory of plane electromagnetic waves,”
Proc.
IRE,
vol. 25, pp. 1457-1493, Nov. 1937
L. J. Chu axial W. L, Barrow, “Electromagnetic
waves in hollow metal
tubes of rectangular cross section,” Proc, ?RE, vol. 26, pp. 1520-1555,
Dec. 1937.
Computations
of Frequencies and Intrinsic Q Factors
of TEO~~ Modes of Dielectric Resonators
JERZY KRUPKA
,4ntract
calculate
E—
Fig. 6.
Transverse
modaJ field distribution
modes),
— The
Rayleigh-Ritz
the resonant
frequencies
method
is described,
and intrinsic
Q factors
which
is used to
due to dielectric
H––-—for a circular
waveguide (first 30
0018-9480/85
Manuscript received August 7, 1984; revised October 29, 1984.
The author is with the Instytut Technologii
Elektronowej,
Politechnika
Warszawska u1. Koszykowa 75, 00-662 Warszawa, Poland.
/0300-0274$01
.00 01985
IEEE