proceedings of the fifth colloquium on … · k betwee~ two cavities and the normalized coupling...
TRANSCRIPT
MICROSTRIP TO WAVEGUIDE COUPLING THROUGH HOLES
Wolfgang J. R. Hoefer* and David S. James**
1. Introduction
A significant limitation encountered in the design of microwave
integrated circuits (MIC's) is that of inadequate Q - factors and poor
temperature stability for currently available MIC reson~tor designs.
It is therefore iesirable that microwave integrated circuits be made
compatible with standard microwave waveguide components and subsystems.
The few transition designs presently available are empirical. This
paper describes a novel practical form of microstrip-to-waveguide transi-
tion whose properties may be readily predicted analytically.
In particular, we have analyzed and measured the coupling between
microstrip (dominant mode) and a circular-cylindrical cavity (TEo12 mode).
The configuration that we have studied has an aperture in the groundplane
of the microstrip. This groundplane also forms one of the walls of the
cavity. The micros trip line is terminated to reflect a short circuit at
the coupling aperture. (See Fig. 1)
1Wheeler's equivalent energy concept for coupling through small holes
forms the basis of analysis. In the experimental study, the coupling1
coefficient S and the loading power factor p (as defined by ~~eeler ) are
determined by measuring the power transfer from the cavity to the micro-
strip and the change in the bandwidth of the system, as a function of the
coupling hole diameter. The cavity is excited by a small loop in the
bottom wall.
**Communications Research Centre, Department of Communications, Ottawa,
Ontario, Canada.
*Department of Electrical Engineering, University of Ottawa, Ottawa,
Ontario, Canada. PROCEEDINGS OF THEFIFTH COLLOQUIUM
ON MICROWAVE COMMUNICATIONMT - 221
Budapest, 24-30 June, 1974
input
microstri p
to cavi ty
coupling
Figure 1 View of the coupling arrangement between
micros trip and TEo12 Cavity (loosely coupled input)
2. Analysis of the Coupling Arrangement
2Wheeler has shown that aperture coupling between a waveguide and a
resonant cavity can be evaluated by combining the solutions of two sym-
metrical coupling problems. The coupling between two cavities each iden-
tical to the resonant cavity in question must be found,as well as the
coupling between two waveguides, each identical to the waveguide in the1
original problem. Using Wheeler's nomenclature the coupling coefficient
k betwee~ two cavities and the normalized coupling reactance x between two
waveguides can be expressed in terms of effective volume ratios. Combina-
tion of one half of each symmetrical structure yields the circuit to be
analyzed (see Fig. 2) and the loading power factor p = l/Q of the1 ext.
cavity is given by
p = k . x + O«k'X)2) (1)
The coupling factor 8 defined by Montgomery4 is related to p, Q ,0
Qext'QL by Q08=-Qext
dp = 8 ..!.-= Q . p an 13+1 QL
0
MT - 222
In our study a circular TEO12 cavity has been chosen for the follow-
ing reasons:
a) It exhibits a maximum unloaded Q for a given ~vity volume. Thus,
critical coupling can be achieved through employmertt of relatively
small apertures.
b) Electrical contact between the main body artd the top and bottom
walls is not required. This guaranties unchanged electrical properties
after repeated assembly and disassembly of the cavity.
c) Magnetic coupling through holes in the top or bottom wall is
easily achieved.
a
...,-1--,
~ ,'
m
'~~.
i
' -"\~', I .-r"o,"
,
'" -
~ T '-:'", '-J I /,
1'0.~-,/
i
i
p
b c-Ft8ure 2
Combination of two
symmetrical coup-
ling problems (b
and c) to obtain
'-t(,((~((((J-'
~ ...,,---
the power loading
factor p of a
cavity coupled
to a microstrip
r t j
:
I
' :
I II I
WI,'!
. I , .
I
line.
-- k x
2.1 AE.e_r~_ureCoupling Between Two. Iden~i2al_TEu 2 Cavities
Since the normal component of electric field of the cylindrical
TEO12 mode is zero on the cavity wall containing the coupling aperture,
only magnetic coupling can be achieved between the ~wo identical resonators1
shown. According to Wheeler, the coupling coefficient k is given by
1k = _4 (V Iv)mc m(2)
MT ...;223
where Vmc is the effective volume of the coupling aperture, and Vm the
effective volume of the cavity. (The effective volume of the cavity !s
defined as the volume which, when filled with a field of uniform intensity
equal to the mag~etic field at the coupling hole, would contain the same
energy as the actual cavity.)
1From~fueeler , for a circular aperture of diameter d:
v = 2 d3/3mc (3)
The effective volume of the cavity is found as follows.
the tangential magnetic field in the cavity at the location of~
ling hole (before the hole was cut), and H the magnetic vector field func-
tion in the cavity at resonance. V is then defined by the expression:, m
~Let H becthe coup-
Thus
1:.~IH 12 . V = 1. ~ (IHI2dV2 c m 2 J I
cavity
JIHI2dVV = cavity
m IH 12c
(4)
(5)
The numerator of Equation (5) is integrated using Lommel's integrals (see2
for exampleAngot). We obtain
f ~ 'lTLR2
)]2
IHI2dv = ---r-' [J 0 (k1R(6)
cavity
where L = Length of the cavity; R = Radius of the caVity; k1 = xo1/R,
where XO1 is' the first root of the equation J1(X) = o.
The magnetic field at the coupling hole is given by
+ kgIH I = - J1 (k1s)
C y(7)
MT - 224
where s is the distance of the aperture from the cavity axis, k3 = 2n/L,
and y = [(XOl/R)2 + (2n/L)2] ~
In order to achieve maximum coupling, the aperture is located at the
maximum of the radial magnetic field.which occurs at s ... 0.481 R. The
coupling coefficient can now be expressed by combining equations (3), (6)
and. (7).
V 1 ,"2 3 k32 [k = 1.1!1.£...4" . '3 d (1) Jl(klS)]2
4 Vm 1TLRZ
Z- [Jo(klR)Y
which reduces to:
(8)
k = 8.7428 d3Ao'[2 L31)2
where Ao is the free space wavelength at resonance.
(9)
2.2 Aperture aoupli~g Between Two Identical Microstrips
Coupling holes in waveguide walls are discontinuities whose effect is
readily described in terms of lumped element equivalents. To simplify the
evaluation of the equivalent impedance of the coupling hole in the common3
groundplane, Wheeler's microstrip model is used. It is an idealized para-
parallel-platewaveguide with magnetic sidewalls. (Fig. 3)
(10 )2
feff = "Ag
Fig.3a Cross-section of a microstrip
line.
y
x
;A =.h.. ~
Zo~~
Fig.3b Equivalent parallel
plate model with mag-
netic sidewalls.
MT -: 225
3
Expressio~s for £eff and6z0 are given by'Wheeler, or more accurately byGetsinger and Schneider; AO and A are the free space and guide wave-glengths, respectively. The TEM fields in the model are simply:
H=H .1x(10)
+ +E = -1;; H jx
(11)
+ +where i and j are unit
~"].l0r; = .
£o£eff
vectors in x and y directions respectively and
In the case of Fig. 2c, the coupling impedance of the hole as seen
from either guide, is the same as that of an identical hole in a common
end wall (see Fig. 4). This can be shown by the following argument. In
the absen~e of the hole, the magnetic field at the endwall (Fig. 4b) is
the same as the field at the bottom wall at odd multiples of A /4 from thegstub. The electric field is zero because the open circuit termination
" ,
reflects a short circuit at the coupling hole and thus the coupling is
purely magnetic.
Z to-.Id
Z JI " i
r- 3A,/4~OPEN-1
CIRCUIT .
TdZ z
1 )~
'Fig. 4a Two identical waveguides
coupled through a hole in
the common bottomwall.
Fig.4b Two identical waveguides
coupled through a hole in
the coman endwall.
4Using the small hole approximation t the reactance of a small
circular hole in the common endwall can be expressed as
7TVx =~ 11 1
"2
2A So ng(12)
MT - 226
...-
where ~ is the transverse component of the unperturbed mode at the loca-
tion of the hole center, assumed uniform across the aperture. V. . 4 ~
is given by Equation (3), and the normalizing factor So is found to be
So = H2 .. A . hx (13)
Thus, the normalized reactance of the iris becomes
.'ITdx=- (14)3A Ahg
2.3 Coupling Between Cavity and Microstrip
The loading power factor of the cavity coupled to the microstrip can
noW be found by combining the results of Equations (9) and (14). We
obtain
l
p = k . x = 0.9276 d6A~L3D2A Ah
g .
(15)
This is the expression presented graphically as a function of the
hole diameter d in Fig. 5. This is easily generalized for finite wall
thickness t and TEOln cavity modes, to
0.2319Id6n2A~ -7.36t/dp = . e
L3d2AhAg
(16)
3. ~£erimental Results
We have measured the loading power factor p and the coupling coeffi~
cient S (the VSWR at resonance) of a TEo12 mode cavity coupled to a 50~
microstrip line. In order to minimize errors due to dimensional inaccura-
cies, measurements were made at a relatively low frequency
using oversize substrates (Rexolite, h - 5 rom, £ = 2.60).rmay be easily applied to ~uch higher frequencies by simply
(e.g. up to 33.4 GHz for h = 0.5 rom).
(3.343 GHz) ,
The result.s
resealing
MT - 227
7Previous measurements by Douville and James yielded an effective
dielectric constant E ff = 2.27 at 3.343 GH for 50~ lines on Rexolitee Z 3
substrates (h = 5 rom,w/h = 2.75). The width A of Wheeler's equivalent
parallel plate waveguide was calculated for h-. 5 mm to be 25 rom,and the
theoretical loading power factor p was calculated (Equation 15).
The groundplane of the microstrip which formed the top wall of the
resonator, was isolated from the body of the cavity by a 1/16" thick
teflon ring to suppress the unwanted TMl12 companion mode. This also
avoided the necessity of providing a reproducible electrical contact
between the cavity walls. The cavity was excited through a small loop in
the bottom wall (Fig. 1). The loading effect of the input was included
into the overall "unloaded Q" of the cavity, which was measured to be
2.31 x 10" :t1%.
The length of the open-circuited matching stub was 3A /4. The end8 g
effect was calculated using the results of James and Tse. The calculated
length of 42.1 mm provided a well-defined peak of coupling over the whole
range of hole diameters investigated in this study. The result is perhaps
somewhat surprising for the large hole diameters. Anomolous behaviour
does occur if the stub length is A /4 rather than 3A /4, SA /4...g g g
The diameter of the coupling hole was successively increased in steps
of about 1 rom, starting at 3 rom, The 3 dB bandwidth and the transmitted
power were measured at the matched output of the microstrip for each
value of d, yielding the loading power factor p and the coupling coeffi-
cient 6. Fig. 5 shows the measured values of the loading power factor
together with it's theoretical behavior (Equation ~). The crosses
represent values obtained with transmitted power measurements. The dots
were obtained from measurement of the bandwidth.
In Fig. 6, the hole diameter is presented as a function of the theo-
retical and measured values of the coupling coefficient. This figure
could be considered as a design chart to determine the coupling hole
diameter for a desired coupling coefficient.
MT - 228
11-]
10-8
+ ,..
t/G
/t/~/
~/
+//-- THEORY
+
: I MEASUREMENTS0
II 2 4 Ii . 18
Hole Diameter in mm
12 14 11
Fig. 5 External loading factor vs. coupling hole diameter for a TEI12
cavity coupled to 50Q microstrip line (E = 2.60; w/h = 2.75;rh - 5 mm; f = 3.343 GH~; L = 13.9 and D = 14.3 ems).
It is surprising how well the measurements agree with theoretical
results obtained from a rather idealized microstrip model and small hole
theory.
The accuracy of the measurements varies with the degree of coupling.
For weak coupling (p < lO-S), transmitted power measurements made with a
resolution of about 0.1 dB yield best results, wherea~ for stronger coup-
ling, measurements of the bandwidth (typically around 300 kHz) made with
an error of tl kHz are more accurate. The diameter of the coupling hole
was measured with an accuracy of about t 1%.
4. Conclusion
Microstrip to cavity coupling has been analyzed using Wheeler's
MT - 229
10-4Q.....eu.af 10-5
2>
110-6-'
. 10-7
6~"t" a
,,~
""~
+"'+>-0::0'wJ:I-
V)I-2w:Ew0::::>V)«w:E
++
0
0
...... ..,..... or........ a.... ..-
CI....
...,Ia....
....
ww U! P JG~.WD!a GIOH
Coupling hole diameter vs. coupling coefficient for a TEo12
cavity coupled to a 50~ microstrip line (8 = 2.60;r
w/h = 2.75; h = 5 rom; f = 3.343 GHz; L = l3.9cms;
D = 14.3 ems).
Fig. 6
equivalent energy concept for coupling through small hol.es. Th~ micro-
strip was represented by an equivalent parallel-plate waveguide. Measure-
ments of the loading power factor and the coupling coefficient agree well
with the theoretical results obtained from this simple model over a wide
range of hole diameters. As an example, the hole diameter, as calculated
for critical coupling between a circular cylindrica~TEo12 cavity and a
50~ micros trip line, is only 2% greater than the measured value. This
percentage corresponds to the estimated overall accuracy of the measure-
ment itself. Microstrip coupling to a waveguide rather than to a cavity
can be readily achieved in the same manner, with a corresponding simplified
analysis.
The analysis presented above may be trivially extended for more
seGeral aperture shape~and other cavity configurations.
MT - 230
-I -a C.... .!!
v--=....GI0
t'4 UID
m....c
a..0
C"I UID....
The coupling arrangement described is especially attractive for
narrowband filter and oscillator applications. It shows great potential
in overcoming several of the more serious problems associated with existing
MITC design techniques.
5. Bibliography ~
~ee1er, H. - Coupling holes between resonant cavities or waveguides
evaluated in cerms of volume ratios, IEEE Trans. on Microwave Theory and
Techniques, Vol~ MTT-12, March 1964, pp. 231 - 244.
2Angot, A. - Complements de Mathematiques, Editions de 1a Revue
d'Optique, Paris, 1965, p. 370.
3Wheeler, H. - Transmission line properties of parallel strips
separated by a dielectric sheed, IEEE Trans. on Microwave Theory and
Techniques, Vol. MTT-13, March 1965, pp. 172 - 185.
4Montgomery, C. G. - Principles of Microwave Measurements, Boston
Technical Publishers, 1964, p. 176.
5Getsinger,W.J. - Microstrip Dispersion Model, Trans. IEEE,
Vol. MTT-21, January 1973,00.34 - 39.
6Schneider, M. V. - Microstrip Dispersion, Proc. IEEE, Vol. 60, No.1,
January 1972,.pp. 144 - 146.
7 .Douville, R. J. P. and James, D. S. - Experimental characterization
of microstrip bends and their frequency dependent behaviour, International
Electrical, Electronics Conf., October 1973, Toronto, Ontario, Canada.
8James, D. S. and Tse, S. H. - Microstrip end effects, Electronics
Letters, Vol. 8, No.2, 1972, pp. 46 - 47.
MT - 231