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ANALYSIS OF A MODIFIED VARIABLE-LENGTH
TUNING POST IN A RECTANGULAR WAVEGUIDE
J. Roelvink and A.G. Williamson
Department of Electrical and Computer Engineering
The University of Auckland
Private Bag 92019, Auckland, New Zealand
[email protected]
A modified variable-length post for use in high power waveguide tuning networks is analyzed
to determine its reactance properties. Theoretical predictions are shown to be in good agreement
with experimental results. An investigation of the reactance characteristics and the aperture voltage
characteristics of the modified post, as a function of its dimensions, is also reported.
Submission Date: 8 December 2006
Acceptance Date: 19 June 2007
Publication Date: 17 August 2007
INTRODUCTION
Impedance matching and filter networks can
be realized in a rectangular waveguide by
introducing reactive discontinuities at certain
points along the waveguide. Over the years, many
waveguide discontinuities have been investigated
and there exists much documented literature on
their equivalent reactance, e.g. [Ragan, 1948;
Montgomery et al., 1948; Marcuvitz, 1951;
Lewin, 1951; Schwinger, 1968; Rizzi, 1988;
Williamson, 1985, Williamson, 1986].
A commonly used waveguide discontinuity
is the variable-length cylindrical post as it is
easy to realize in practice and can be tuned
over a relatively large capacitive range. The
variable-length adjustment is usually achieved
by using a threaded post to enter the waveguide
through a threaded hole (often with a lock-nut to
Keywords: Waveguides, waveguide discontinuities,
impedance matching networks
41-2-12
prevent subsequent movement). A shortcoming
of this approach is the limited metal-to-metal
contact in the threaded section that results in
a small but non-zero contact resistance. Whilst
this form of construction may be satisfactory in
many situations, for those where it is not, such
as in high power applications, an alternative
geometry is required.
A technique to reduce the effect of the contact
resistance associated with the threaded section
is the introduction of a coaxial choke, shown in
Figure 1(a), which presents a low impedance at
the coax/waveguide aperture that is relatively
insensitive to small changes in the resistance
due to the screw thread (in practice there would
be a tight fit between the post and the choke,
which could easily be supplemented by an
external clamp-ring after post adjustment). The
post remains easily tunable by the threaded
section beyond the choke and external to the
waveguide. For a given frequency the effect
of the coaxial choke on the fields within the
Journal of Microwave Power & Electromagnetic Energy
Vol. 41, No. 2, 2007
h
z
Coaxial choke
section
Screw thread
2a
�
y
x
2b
e
(a)
� OFF
d
(b)
Figure 1. Modified variable-length tuning post (a) sectional view (b) simplified
sectional view.
waveguide can be modeled by the simplified
arrangement shown in Figure 1(b) where
there is a coaxial line section offset from the
coax/waveguide aperture by a fixed length lOFF
(more complicated choke arrangements can be
modeled in an equivalent form of Figure 1(b)).
In some practical implementations it has been
noted that for some equivalent values of lOFF
the reactive effect of the modified post can be
rather insensitive to insertion depth, l. Of course,
when lOFF is a multiple of λ/2, the reactive
characteristic of the modified post should be the
same as the standard variable-length post, that
is with lOFF =0. Clearly, if this more expensive
design to manufacture is to be used in a high
power application to overcome the limitations
of the standard variable-length post, a theoretical
analysis or model that accounts for the effect of
the coaxial choke would be invaluable to assist
in determining appropriate dimensions. This is
the motivation for the work reported here.
In this paper an accurate theoretical analysis
for the modified tuning post is presented.
Theoretical and experimental results are reported
that both confirm the accuracy of the model and
illustrate examples of where the reactance can
International Microwave Power Institute
be quite insensitive to post depth as observed in
practice. The reactance characteristics and the
voltage in the coaxial aperture and coaxial line
section are investigated and reported, illustrating
what values of lOFF produce insensitive reactance
characteristics and relatively large aperture
voltages.
THEORY
The modified tuning post in Figure 1(a) is
considered here as the simplified configuration
shown in Figure 1(b) where there is a cylindrical
post entering through a coaxial aperture that is
shorted at an offset length, lOFF , and extends into
the waveguide a length l (the coordinate system
adopted here is that associated with the coaxial
line). It is assumed that the waveguide and
coax dimensions, and the operating frequency
are such that only the TE10 mode can propagate
in the waveguide, and only the TEM mode
can propagate in the coaxial region. All metal
surfaces are modeled to be perfectly conducting
and to simplify the analysis it is assumed that the
fields in the vicinity of the post are rotationally
symmetric as in [Williamson, 1985].
41-2-13
The analysis involves finding the induced
current distribution on the post surface resulting
from the incident TE10 mode field, namely:
E
inc
z
(
 π x  − jk y
x, y,z = Ei sin   e 10
 d 
)
(1)
where k10 = k 2 − ( π d ) , k=2π/ λ and Ei is the
magnitude of the incident TE10 mode field.
Once the induced current has been determined,
the scattered field and resulting reactive effect
can be calculated by a similar method to that
in [Williamson, 1986]. The calculation of the
induced current is facilitated by the extension
of two existing analyses, namely the induced
current on a variable-length coaxially driven
hollow post, IC (z) [Williamson, 1985] and the
induced current by an incident TE10 field on a
variable-length hollow post, IH (z) [Williamson,
1986]. These currents were obtained from the
solutions of two integral equations and the
details of their evaluation can be found in
[Williamson, 1986].
Upon applying the principle of superposition,
an expression for the total current induced on the
post shown in Figure 1(b) can be given by:
2
()
()
()
IT z = Ei I H z + V I C z
(2)
where V is the voltage in the coax/waveguide
aperture at z=0. An obvious difficulty in the
evaluation of (2) arises in determining V as the
form of the electric field if the aperture is not
known. However, if the aperture electric field
is assumed to be of the same form as the TEM
mode in the coaxial stub, the post is electrically
thin (ka<<1) and kl>>ka. A good approximation
for V can be given by:
()
V = − jX AP IT 0
(3)
where IT(0) is the current flowing on the post
at the coax/waveguide aperture and XAP is the
41-2-14
reactance of the shorted coaxial section referred
to the aperture, namely:
(
X AP = ZC tan klOFF
)
(4)
where ZC is the characteristic impedance of the
coaxial region (note that for more complicated
choke designs the expression for XAP will be
different).
By considering the total current at z=0, the
voltage V can be found from (2) and (3). It
is then a simple matter to calculate the total
current from (2). The scattered TE10 mode
field from the current on the post and the field
in the aperture (using the TEM approximation
[Williamson, 1985]) can be calculated using a
similar approach to that adopted in [Williamson,
1986] and shown to be:
 π x  − jk y
Ezscat x , y , z = Er sin   e 10
 d 
(
)
(5)
where
 e  
2 jV
Er = sin   
 d   h ln b a k10 d
( )

 J kb − J ka  − kη0 I J ka 
0
 0
 k d 0 0
 (6)
10
( )
( )
( )
and I0 is the z-independent part of IT(z), η0 is the
intrinsic impedance of free space while J0 is a
Bessel function of the first kind. The first term of
(6) represents the scattered TE10 mode field due
to the coaxial line section while the second term
represents the scattered TE10 mode field by the
total current on the post. By comparing (1) and
(5), it can be seen that the reflection coefficient
is given by:
E
R= r
(7)
Ei
The effect of the modified variable-length tuning
post can be represented by the equivalent circuit
shown in Figure 2 containing a single normalized
Journal of Microwave Power & Electromagnetic Energy
Vol. 41, No. 2, 2007
shunt reactance (a good approximation if the
post is electrically thin), where R and X are
related by:
R=−
1
1 + 2 jX
(8)
COMPARISON OF THEORETICAL AND
EXPERIMENTAL VALUES FOR REACTANCE
The normalized reactance of a 6 mm diameter
post (2a=6 mm) with a 15 mm diameter aperture
(2b=15 mm) in a C-Band waveguide (d=47.55
d
d=47.55
mm, h=22.15) was measured over the usual
operating frequency range of the guide (3.955.85 GHz) for several fixed offset lengths, lOFF,
using a similar approach to that outlined in
[Roelvink and Williamson, 2005]1.
Theoretical and experimental results for the
normalized post reactance, X, are shown in Figure
3 for two situations lOFF/ λ=0.65 and lOFF/ λ=0.70.
In both cases the agreement between theory and
experiment is very satisfactory. However, the
extensive range over which the post reactance
is relatively unchanged is evident.
CHARACTERISTICS OF THE MODIFIED
TUNING POST
In Figure 4, the normalized reactance, X , has
been plotted as a function of l/h for a number
of values of lOFF. It can be seen that when lOFF
is in the range 0 < lOFF/λ < 0.25, the reactance
is greater than that of the standard tuning post
(lOFF=0) while in the range 0.25≤ lOFF/λ< 0.5
the reactance is of a lesser value. Note that the
reactance changes widely when lOFF/λ is close
to 0.25 as in this case XAP is large.
An important parameter to consider,
Waveguide
Port 1
jX
Waveguide
Port 2
Figure 2. Modified variable-length
tuning post equivalent circuit.
particularly in high power applications, is the
induced voltage in the aperture, V . Once V has
been determined, all voltages in the coaxial
choke can be determined. Figure 5 gives results
for V
V, calculated from (2) and (3) (with Ei=1),
as a function of l/h for several values of lOFF. It
can be seen that as lOFF increases from zero, the
voltage in the aperture increases. When lOFF/
λ=0.2 there is a significant voltage increase
around l/h=0.10. In general, the trend is for the
voltage to be largest around lOFF/λ=0.25 when
XAP is large (for this case this is also the location of
the maximum voltage in the coaxial line region).
As lOFF/λ is increased beyond 0.25 towards 0.5,
the aperture voltage decreases (the maximum
voltage now occurs in the coaxial line section).
The aperture voltage is of interest as the corners
associated with the transition from coaxial line
to rectangular waveguide give rise to relatively
high electric fields and so arcing commonly
occurs first in this region. Consequently, values
of lOFF resulting in relatively high aperture
voltages should be avoided.
In Figure 6 the reactance is shown for three
frequencies when lOFF/λ= 0.2. It was found that
the difference in reactance between each of the
frequencies was relatively larger for values
of lOFF/λ that are close to 0.25 than for values
around 0.5.
The above results illustrate that while it may
be possible to increase the reactance tuning range
for some values of lOFF, the disadvantages of
1
In these measurements a solid post was used, whereas the theory presented here is for hollow ended posts. The
difference in reactance of each form has been investigated elsewhere [Roelvink and Williamson, 2005]. In order to
invoke a more meaningful comparison, the effective hollow length, ls, was taken to be
ls=l + 0.12a [Roelvink and Williamson, 2005].
International Microwave Power Institute
41-2-15
4
�OFF = 0.70�
X
2
0.65�
0
-2
-4
0
0.2
0.4
�/h
0.6
0.8
1
Figure 3. Theoretical and experimental results for the normalized reactance, X, for several
values of lOFF as a function of l/h for the case h=22.15 mm , d=47.55 mm , e/d=0.5, 2a=6 mm,
2b=15 mm and f=4.9 GHz.
— Theory, ••• Experimental results.
4
�OFF = 0.2�
0.1�
0
X
0�
0.4�
-4
0.3�
0.25�
-8
0
0.2
0.4
�/h
0.6
0.8
1
Figure 4. Theoretical results for the normalized reactance, X, for several values of lOFF as
a function of l/h for the case h=22.15 mm , d=47.55 mm , e/d=0.5, 2a=6 mm, 2b=10 mm and
f=4.9 GHz.
insertion insensitivity, increased aperture voltage
and frequency sensitivity are significant. A value
of lOFF somewhere in the region 0.35λ−0.6λ
produces a reactance range similar to that of
the standard variable-length tuning post that is
relatively sensitive to insertion without greatly
suffering from the above disadvantages.
41-2-16
CONCLUSION
A modified variable-length tuning post for use
in high power impedance matching applications
has been presented and an accurate analysis for
predicting it’s reactance characteristics has been
developed that has been shown to be in good
agreement with experimental measurements.
Journal of Microwave Power & Electromagnetic Energy
Vol. 41, No. 2, 2007
10
0.015
5.1GHz
5
f = 4.9GHz
V
X
�OFF = 0.2�
0.01
0.1�
0.3�
0.005
-5
0.4�
0
0
4.7GHz
0
0.2
0.4
�/h
0.6
0.8
1
-10
0
0.2
0.4
�/h
0.6
0.8
1
Figure 5. Theoretical results for the
aperture voltage,V, for several values of lOFF
as a function of l/h for the case h=22.15 mm ,
d=47.55 mm , e/d=0.5, 2a=6 mm, 2b=10 mm
and f=4.9 GHz.
Figure 6. Theoretical results for
the normalized reactance, X, for several
frequencies as a function of l/h for the case
h=22.15 mm , d=47.55 mm, e/d=0.5, 2a=6
mm, 2b=10 mm and lOFF=0.2λ.
This model has permitted an investigation into
the reactance and voltage characteristics of the
modified tuning post. A choice of 0.35<lOFF/
λ<0.6 would produce a tuning post with a
useful tuning range. Developing a simple
tuning structure that can be tuned through both
a capacitive and inductive range is the focus of
ongoing research by the authors.
Lewin, L. (1951). Advanced Theory of Waveguides.
Waveguides
Iliffe, London.
Schwinger, J.S., and Saxon, D.S. (1968). Discontinuities
in Waveguides: Notes on Lectures.
Lectures Gordon and
Breach, New York.
Rizzi, P.A. (1988). Microwave Engineering Passive
Circuits. Prentice-Hall, New Jersey.
Williamson, A.G. (1985). “Coaxially fed, hollow probe in
a rectangular waveguidE.” IEE Proc. H, Microwaves,
Antennas & Propag. 132, pp.273-285.
Williamson, A.G. (1986). “Variable-length cylindrical
post in a rectangular waveguide.” IEE Proc. H,
Microwaves, Antennas & Propag. 133, pp.1-9.
Roelvink, J., and Williamson, A.G. (2005). “Reactance of
hollow solid and hemispherical-cap cylindrical posts
in rectangular waveguide.” IEEE Trans. Microwave
Theory Tech. MTT-53, pp.3156-3160.
ACKNOWLEDGMENTS
The authors wish to thank Keam Holdem
Associates, Auckland, New Zealand, for being
the industrial partner of a Technology New
Zealand TIF Award, and Dr. R. Keam for several
helpful discussions.
REFERENCES
Ragan, G.L. (1948). Microwave Transmission Circuits.
McGraw-Hill.
Montgomery, C.G., Dicke, R.H., and Purcell, E.M. (1948).
Principles of Microwave Circuits. McGraw-Hill.
Marcuvitz, N. (1951). Waveguide Handbook . McGrawHill, New York.
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