MICROWAVE ELECTRONICALLY TUNABLE FILTERS
by
IAN CHARLES HUNTER B .Sc
A Thesis presented for the degree of
Doctor of Philosophy, in the
Department of Electrical and Electronic Engineering
at the University of Leeds
May 1981
ACKNOWLEDGEMENTS
The author wishes to thank Professor J.D. Rhodes
for his expert advice and supervision throughout the period
of this research.
The author would also like to thank
Professor P.J. Lawrenson for permission to undertake the
research in the Department of Electrical and Electronic
Engineering at the University of Leeds and for the use of
the departmental facilities.
Finally, the author wishes to
thank Filtronic Components Ltd., Ferranti Ltd. and the Science
Research Council for technical and financial support.
ABSTRACT
Design procedures are presented for varactor tuned
microwave bandpass and bandstop filters constructed in
Microwave Integrated Circuit form.
The tunable bandpass
filters were based on a combline filter incorporating novel
input and output coupling networks which compensate for the
frequency dependence of the coupling between the resonators.
Using this filter, tuning over an octave band with an accept­
able response and approximately constant passband bandwidth
is possible.
The tunable bandstop filters consisted of a
uniform impedance
main line with capacitively decoupled
resonators located at intervals along it.
A novel design
technique is presented for evaluating the correct phase
shifts between the resonators of this filter in order to
obtain the
optimum symmetrical frequency response. Detailed
computer analysis of the filters,
including both varactor
and integrated circuit loss, and the effects of tuning is
presented.
The measured performances of several practical
devices, constructed in Suspended Substrate Stripline, and
operating in the frequency range 2-10 GHz are presented,
these agree closely with computed performances.
MICROWAVE ELECTRONICALLY TUNABLE FILTERS
Contents
Page No.
Chapter 1
-
Introduction
1
1.1
Applications of Tunable Microwave Filters
1
1.2
YIG Filters
1
1.3
Varactor Tuned Filters
2
1.4
Choice of a Suitable Construction Medium
for the Microwave Filters
5
1.5
Development of Microwave Filters Suitable
for Varactor Tuning
8
References
10
Initial Investigations of Tunable Microwave Filters
12
2.1
Tunable Bandstop Resonators
12
2.2
Tunable Bandstop Filters
22
2.3
Tunable Bandpass Filters
32
2.4
Conclusions
34
References
35
Appendix 2.1
Getsingers Curves for Coupled Rectangular
Bars
36
Chapter 3
Tunable Microwave Bandstop Filters
40
3.1
Introduction
40
3.2
Theory of Symmetrical Bandstop Filters
41
3.3
Design of fixed frequency Bandstop Filters
3.4
Varactor Tuned Bandstop Filters
58
3.5
Large Signal Effects
66
3.6
Conclusions
69
References
69
Chapter 2
-
-
49
Page
Chapter 4
-
Tunable Microwave Bandpass Filters
71
4.1
Introduction
71
4.2
Theoretical design of Tunable Microwave
Bandpass Filters
72
4.3
Physical design of Tunable Microwave
Bandpass Filters
91
4.4
Conclusions
108
References
109
Appendix 4.1
A derivation of the values of the transformer elements of the Tunable Combline
Filter
110
Appendix 4.2
A comparison of the Q factors of
varactor tuned combline and Open-Circuited
Combline Filters
112
Chapter 5
General conclusions
115
1
1
INTRODUCTION
1.1
Applications of Electronically Tunable Microwave Filters
There is a large and growing demand for frequency agile
microwave filters for
Measure
application in Electronic Support
(E.S.M) systems.
For example, a tunable bandpass filter
can be used in a crystal video receiver [1 .1 ] to provide
frequency discrimination of unknown signals
[Fig. 1.1.1].
Tunable bandstop filters have wide application in similar
systems for the purpose of rejecting unknown interference
signals.
E.S.M systems typically operate over multi-octave
frequency bands and thus the first requirement for tunable
microwave filters is that of broadband tuning.
In addition,
in order that relatively fine frequency discrimination can be
achieved, the filter passband
1.2
(stopband) must be narrow.
YIG Tuned Filters
At present, most E.S.M system requirements are met
only by using YIG filters [1.2].
The YIG filter is composed
of Yittrium iron garnet spheres located between the poles of
a dc magnet.
The YIG spheres resonate in the microwave spectrum
and variation in the magnetic field changes their resonant
frequency.
YIG filters can be tuned over greater than octave bands
and the resonator Q factors are in excess of 10000.
Unfortun­
ately their inherant magnetic hysteresis effects result
in
low tuning speeds, typically of the order of one microsecond
per Megahertz.
<
Tunable
bandstop
filter
Tunable
bandpass
filter
6
■O
\/
crystal
detector
amplifiei
Fig. 1.1.1
A Tunable Video Crystal Receiver
2
1. 3
Varactor Tuned Filters
1.3.1
Fundamentals of Varactor Diodes
For certain E.S.M systems operating in a dense pulse
environment it is desirable to have filters with the capabil­
ity to tune across broad bandwidths at speeds in excess of
one microsecond per Gigahertz.
The use of filters without
this capability would result in a degradation of the receivers
probability of intercept.
To achieve such high tuning speeds, or slew rates, it
is necessary to use varactor diodes as the tuning elements
for microwave resonators.
Varactors essentially consist of
P.N. junctions operated in reverse bias.
A variation in the
reverse bias voltage will alter the width of the junction
depletion region, thus changing the diode capacitance.
The
response time of a typical microwave varactor to a step change
in bias voltage is of the order of one picosecond.
suitable design of the varactor bias circuit,
Thus by
it is possible
to achieve the required high speed tuning.
The details of the physical analysis and construction
of varactor diodes are beyond the scope of this thesis, only
their relevant properties are presented here.
information the
reader is referred to ref.
For further
1.3.
The most commonly used form of tuning varactor is the
abrupt junction varactor.
This consists of a P+N
junction
diode with an abrupt change in doping density at the junction
surface.
The doping density of the N
epitaxial layer region
is constant throughout its width and depletion occurs mainly
in this region [Fig. 1.3.1].
The voltage dependence of the
diode capacitance is given by
Co
Cj (v)
(1
+
(1.3.1)
v/\p)
where:
Co
A
e
=
=
(1.3.2)
Area of the diode
Dielectric constant of the epitaxial layer material
e
=
The electronic charge
Nd
=
Epitaxial layer doping density
V
=
Reverse bias voltage
\p
=
Built in junction potential
The maximum capacitance variation is limited by the
diode breakdown voltage which can be up to sixty volts.
Thus
an 8:1 capacitance variation is possible from a varactor chip.
With the voltage capacitance variation given by 1.3.1
the voltage frequency relationship of the resonance of an LC
circuit using a varactor as the capacitor is given by
(1.3.3)
This highly non-linear tuning characteristic can be
compensated for by using varactors with other doping profiles,
e.g. the hyper-abrupt junction varactor [1.4].
Unfortunately
these diodes suffer from lower Q factors than abrupt junction
varactors and consequently were not used in the work described
in this thesis.
Boundaries of
depletion region
Fig.
1.3.1
Doping profile of abrupt P.N junction
Cj (v)
Rj (v)
■AA/V
Fig.
1.3.2
Small signal reverse bias model of a varactor
4
1.3.2
A Simple Circuit Model for
a Varactor
Assuming that the diode is unpackaged and that the
depletion region is completely depleted of carriers, then the
diode can be represented by the small signal reverse bias
circuit model shown in Fig.
1.3.2.
Since the P+ region in
the diode is very heavily doped, the resistance Rj (v) in the
circuit model is dominantly controlled by the resistance of
the undepleted region of the epilayer.
The
Q factor of the diode can be shown to be given by
the following expression [1.5]
Q
=
2 ttF C j tv) Rj (v)
(1.3.4)
where F is the frequency of operation.
It is obvious that for high frequency operation the
varactor epilayer resistance must be minimised.
This could
be achieved by increasing the area of the diode, however that
would increase the junction capacitance and the doping density
would then have to be reduced, which in turn would increase
the resistance.
To achieve this objective varactors are now
constructed from Gallium Arsenide [GaAs].
This material has
the advantage over Silicon of an approximately seven times
greater electron mobility for a given doping density.
Since
the dielectric constants of the two materials are very similar,
Rj(v) will be seven times lower when using GaAs, as opposed to
Silicon varactors of the same capacitance.
5
1.3.3
Performance of Available Varactor Tuned Filters
Varactor tuned filters are not available for operating
frequencies above 2GHz and a typical specification of a
commercially available U.H.F tunable bandpass filter manu­
factured in the U.S.A [1.6] is shown in Fig.
1.3.3.
If the assumption is made that the midband dissipation
loss of this filter were due only to the varactors,
it is
possible to evaluate the loss of filters operating at higher
frequencies.
For example, a filter with the same percentage
bandwidth and number of resonators as that described in Fig.
1.3.3, but designed to tune from 4GHz to 8GHz, would have a
minimum passband loss of approximately 30dB.
This figure
would be unacceptable for most applications.
The filter
described, was however, constructed using Silicon varactors
and it was anticipated that by using GaAs devices, broadband
tunable low loss filters would be feasible for use at frequenc­
ies up to 10GHz.
1.4
Choice of a Suitable Construction Medium for Varactor
Tuned Microwave Filters
Initial work at Ferranti Ltd.
[1.7] concentrated on
the development of X band tunable waveguide bandpass filters.
The tuning bandwidths of these filters were limited to less
than 10%.
This was caused by the swamping of the varactor
diodes by the parasitic reactances associated with waveguide
discontinuities.
Another disadvantage of these filters was
their relatively complex mechanical construction.
540-1026 MHz
Tuning range
3dB Passband bandwidth
5%
Number of resonators
2
Passband loss
7 dB Maximum
Passband V.S.W.R
2.5:1 Maximum
Tuning signal
0 — 60V D.C.
Intercept point
Fig. 1.3.3
(2nd ordered)
5 dBm minimum
A typical specification for a commercially
available varactor tuned bandpass filter
7
For the above reasons, waveguide filters were abandoned,
as were coaxial filters.
It was thus envisaged that a printed
circuit realisation was the best alternative.
Initially,
microstrip was investigated, but this medium was abandoned
because of its low Q factor and its inability to operate over
broad bandwidths because of spurious mode propogation.
conventional stripline was also ruled out because of its high
dielectric loss.
Previous work on broadband filters and multiplexers
[1.8] suggested that Suspended Substrate Stripline
(S.S.S) was
an excellent medium for the realisation of high performance
microwave filters.
The S.S.S structure consists of a thin
copper clad dielectric substrate suspended midway between the
ground planes of an aluminium housing [Fig. 1.4.1].
the dielectric is thin,
Because
(normally 0.005" or 0.01") propogation
is mainly in air and a loss performance similar to coaxial
circuits of the same volume can be achieved.
Spurious modes
can easily be supressed in this structure and thus broadband
operation is possible.
The vibrational requirements for
many
applications can be met using S.S.S because, with the substrate
located midway between the ground planes, the first ordered
sensitivity of the circuit elements to vibration is zero.
Although this structure is printed,
it is possible to fine tune
filters using capacitive tuning screws located through the main
housing.
In conclusion, S.S.S possesses most of the advantages
of the coaxial and microstrip structures, but few of their
disadvantages.
Aluminium
housing
air filled cavity
R.F
connector
Dielectric circuit
board
Fig. 1.4.1
The Suspended Substrate Stripline structure
8
1.5
Development of S.S.S Microwave Filters Suitable for
Varactor Tuning
1.5.1
Tunable Bandstop Filters
The S.S.S realisation of the tunable bandstop filter
is shown in Fig.
1.5.1.
This filter is composed of a uniform
impedance main line with resonant capacitively coupled stubs
located at intervals along the line.
Varactors can be
connected to the ends of the stubs to provide electronic
tuning.
This filter has the advantages of maintaining a good
passband from d.c to approximately twice the stopband centre
frequency, and the capability of producing very narrow stop­
bands with high rejection.
In chapter two it is shown that coupling the bandstop
resonators via the standard quarter wave impedance inverting
lengths of line results in a non optimum assymetric frequency
response.
This is caused by a fundamental property of the
resonators which each have transmission poles at frequencies
very close to their transmission zeros.
In chapter three a
technique is constructed for evaluating the correct phase shift
between the resonators to compensate for the assymetric
frequency response.
Several bandstop filters have been designed and
constructed and the measured performances of three are presented.
These include two fixed frequency and one varactor tuned filter.
The fixed frequency filters were both 7 cavity devices operat­
ing at 4GHz with respective 40dB stopbands of 1% and 10%.
The varactor tuned filter was a 3 cavity device with a 1% 20dB
Capacitive
Resonator stub
Fig.
1.5.1
Tuning capacitor
The Suspended Substrate Stripline realisation
of
a tunable microwave bandstop filter
9
stopband bandwidth at 4GHz.
This filter was designed to tune
from 3.5GHz to 4.5GHz.
All three devices worked well and the measured perfor­
mances were in excellent agreement with computer analysis.
1.5.2
Tunable Bandpass Filters
It is shown in chapter two that in order to produce a
tunable bandpass filter with a good response over a broad
tuning bandwidth it is necessary to choose a circuit in which
the coupling between the resonators is not strongly dependent
on frequency.
If this were not the case, then the filter
response would only be correct at one frequency and it would
deteriorate rapidly with tuning.
In chapter four, a design technique is presented for
tunable combline filters in which the redundant transformer
elements are used to approximately compensate for the frequency
variation of the coupling between the resonators.
Using this
technique it is possible to tune over an octave band with only
a small deterioration in the filter response.
This type of
filter also possesses the additional important property of
maintaining approximately constant passband bandwidth independ­
ant of tuned frequency.
The theory of tunable combline filters is extended to
the design of combline filters with open circuited resonators.
These filters have a higher Q factor than the normal varactor
tuned combline filter when designed for the same degree and
passband bandwidth and they also avoid the problem of lossy
10
short circuits which becomes noticable when very narrow
bandwidths are required.
The disadvantage of the open
circuited varactor tuned combline filter is a reduced tuning
bandwidth as compared to the normal combline filter.
The design and measured performances are presented
for two varactor tuned bandpass filters.
These include a 2
resonator combline filter designed to tune from 3GHz to 5GHz
with a 5% passband bandwidth, and a 3 resonator open circuited
combline filter tunable from 3.5GHz to 4.5GHz with a 3% pass­
band bandwidth.
Measured performances are in close agreement
with computer anlaysis.
Finally, the design process for a fixed frequency
S.S.S bandpass filter is presented.
This filter, which is
suitable for passband bandwidths of less than 30% is a modifi­
cation of the tunable combline filter such that it requires
no lumped capacitors.
and constructed.
Two of these filters have been designed
The filters were both 3 resonator devices
operating at 4GHz with respective passband bandwidths of 1%
and 5%.
The measured performances of these devices are similar
to those using coaxial circuits.
References
1.1
'Wideband E.S.M. receiving systems part 1'
C.B. Hofmann and A.R.
Baron
Microwave Journal Vol. 23 No 9 Sept 1980
1.2
'YIG filters aid wide open receivers'
W.J. Keane
Microwaves
Sept. 1978 Vol 17 no 8
11
1.3
'Variable Impedance Devices' Chapter 1
Edited by M.J. Howes and D.V. Morgan.
Wiley
1.4
'Circuit theory'
J.O. Scanlan
Vol 1 pp 144-150
and R. Levy
Oliver and Boyd
1.5
'Hyperabrupt varactors from Vapor Phase epitaxial
Gallium Arsenide'
J.L. Heaton and R.E. Walline
New Product release.
Microwave Associates Ltd.
Burlington, Mass.
1.6
'Voltage Tuned Filters *
Microwaves Magazine
1.7
January 1976
Technical Report on Varactor tuned filters
Scott Wilson
Ferranti Ltd. Dundee
l.'B
'M.I.C. Broadband filters and Contiguous Multiplexers'
J.E. Dean and J.D. Rhodes
Proceedings of the 9th European Microwave Conference.
Brighton 1979
12
2
INITIAL INVESTIGATIONS OF VARACTOR TUNED MICROWAVE
FILTERS
2.1
Tunable Microwave Bandstop Resonators
2.1.1 Choice of a Suitable Resonator
The main requirements for a tunable bandstop resonator
were that a good
passband could be maintained over a broad
bandwidth, that narrow stopbands could be achieved and that
varactor diodes could easily be incorporated into the circuit.
These requirements necessitated the use of a straight through
line to achieve a good passband performance, and also a means
of decoupling the resonator from the main line to achieve the
required narrow stopband bandwidths.
These conditions can
be satisfied by the use of many types of coupled line circuits
but the final choice of resonator was the relatively simple
structure shown in Fig.
2.1.1.
This resonator consists of a
transmission line terminated in a variable capacitor at one
end with the other
end capacitively coupled to a main line.
The variable capacitor can be either a varactor, or a tuning
screw
located through the S.S.S. housing.
One very important
advantage of using this resonator is that when multisection
filters are constructed, consisting of resonators coupled at
intervals along a main line then the resonators can all have
the same physical length.
This is because large variations
in the resonator bandwidth can be achieved by varying the
capacitive gap from the resonator to the main line, with only
a small change in the resonant frequency.
Fig.
2.1.1
The S.S.S structure of a tunable bandstop
resonator
C2
Fig. 2.1.2
The equivalent circuit of the bandstop
resonator
13
2.1.2
Design of the Bandstop Resonators
Several bandstop resonators were constructed firstly
in order to obtain information about the effects of the
finite S.S.S. Q factor on narrowband filter performance,
and secondly to determine the exact equivalent circuit of
the resonators.
In order to minimise the number of S.S.S
housings needed, the resonators were all to have the same
length but would have different coupling gaps to the main
line.
The resonators were to operate at approximately 5 GHz,
this being the approximate frequency at which the varactor
tuned filters were eventually to be constructed.
Design of resonator lengths
The equivalent circuit of the bandstop resonator is
shown in Fig. 2.1.2.
The input impedance of this resonator
can easily be shown to be
Zin(jw) =
Where
(with
= 0)
- j[wGZo + tan (aw) ]
wC tan (aw)
,_ , ,
’a' is the propogation constant of the resonator stub.
At the stopband resonant frequency w q we have
L
C
a
w
o
tan 1 [-w CiZ ]
o ^o
where
L
=
Length of the resonator
C
=
Velocity of light in air.
(2 .1 .2 )
14
To design for a specific resonant frequency it is
strictly necessary to be able to relate the value of
to
the dimensions of the coupling gap to the main line.
How­
ever for narrow bandwidths C^will be small, as
tends to
zero we have
a
=
IT
—
wo
Thus for narrow bandwidths the resonator phase
length will approach 180°.
In fact the resonator phase
lengths were chosen to be 170° at
5GHz
which corresponds to
a physical length of approximately 1 .1 ".
Design of widths of S.S.S. transmission lines
The characteristic impedance of an isolated S.S.S.
transmission line is related to its static capacitance
'C'
per unit length to ground by the following expression.
/er Zo
=
n/C/e
(2.1.3)
where
n
=
Characteristic impedance of free space
e
=
Dielectric constant of the S.S.S structure
=
377 fi
eo
The relationship between the dimensions of an isolated
strip transmission line [Fig. 2.1.3] and its static capacitance
per unit length is
...
<r
■»
C f '^T-
—
CP
Cf
--------
>
c
f
1
C
f
1
■>
<r
w
Fig. 2.1.3
An isolated rectangular bar between infinite
parallel ground planes
C-------------7
V
V
C
f
uu
..........
'
i
“
-
T
~
^
C£
\
0
1
^
k—
Fig.
2.1.4
w
C
f
'
/
1
\
oo
—V
A rectangular bar between parallel ground
planes and in proximity to a side wall
15
c
e
=
2Cp
+
4 Cf '
e
(2.1.4)
where Cp is the parallel plate capacitance to ground from
each side of the stripline bar and C f ' is the fringing
capacitance from one corner and half the associated side
wall of the bar to ground.
Getsinger
[2.1] has evaluated
Cf' as a function of bar thickness, this is shown graphically
in Appendix 2.1.
Examining Fig. 2.1.3 we see that
From 2.1.4 and 2.1.5 we obtain
»
b
=
0 .25[—
e
-
(2 .1 .6 )
e
Substituting 2.1.3 into 2.1.6 we obtain
w
g
=
377
0.25[^
-
4Cf'
^
]
(2.1.7)
The width of bar required to realise any desired
impedance transmission line can be calculated using 2.1.7.
Both the main line and the resonator were designed
to have 50 fi impedance.
The ground plane spacing
'b' was
chosen to be 0.15", this would give an unloaded resonator Q
factor of approximately 300 at 5 GHz
The conductor thickness
board was 0.00085", thus
[2.2],
't1 of the S.S.S circuit
16
£
13
=
0.00566
Using this value of t/b we obtain from Appendix 2.1 that
— ' =
e
0.46
From 2.1.7 the resonator and main line widths wl and w2
can be obtained,
wl
these are
=
w2
=
0.213"
The possibility of propogation of unwanted modes
The overall width of the interior of the S.S.S
housing must
be at least equal to the width of the main
line plus the length of the resonator, i.e. the width will
be greater than 1.3".
With this width of cavity the cut off
and resonant frequencies of the lowest possible TE^Q -^ mode
passing through the cavity will be 4. 5 GHz
[2.3]
and 9.06 GHz.
It is thus impossible to stop the propogation of
the TE10l mode at the designed frequency of 5 GHz , however,
the method of launching onto the stripline circuit is
dissimilar to that used in waveguides, also the characteristic
impedance of the S.S.S cavity presented to the T E ^ ^ mode
will be relatively high.
These two factors combined, ensure
that no box effects will be noticed below the resonant
frequency.
In order to minimise the width of the S.S.S cavity,
the side wall was to be located at a distance of 0.05" from
17
the main
line.
The width of the main line must thus be
recalculated to compensate for the increased fringing
capacitance to the side wall.
The capacitance to ground
per unit length from the main line is now
C
£
_
4w2
b
+
M l
e
+
e
Where, Cf'o is the odd mode fringing capacitance from one
corner and half the associated side wall.
This situation is
shown in Fig. 2.1.4.
The odd mode fringing capacitance from a pair of
coupled bars to ground is defined as the fringing capacitance
to ground from each bar when a short circuit is placed midway
between the bars.
Calculation of the odd mode fringing
capacitance of one bar near a side wall is equivalent to that
for coupled bars except that the distance between the coupled
bars is twice that of the single bar to the side wall.
Cf'o
is shown graphically in Appendix 2.1 as a function of bar
separation and bar thickness.
With the main line at a distance of 0.05" from the
side wall we have
s
=
0 .1 "
and
From appendix 2.1 we obtain
Cf'o
=
0. 52
18
Thus, from 2.1.8 the main line width is
w2
=
0.209"
Choice of Resonator coupling gap
No information was yet available for relating the
dimensions of the coupling gap between the resonator and
the main line to its lumped capacitance.
Consequently five
circuit boards were constructed with coupling gaps of between
0.004" and 0.020".
Although these values seem somewhat
arbitrary, it was expected that they would realise relatively
narrow band resonances.
boards is shown in Fig.
The layout of one of the circuit
2.1.5
Design of S.S.S housing and feed terminals
The S.S.S. housing was a relatively simple structure,
built in two halves, with milling in each half to the depth
of one half of the ground plane spacing.
In addition, a
0.05" wide 0.005" deep grove was milled around the edge of
the cavity in the lower housing to clamp the circuit board
into its correct location [Fig. 2.1.6]
In order to produce a low V.S.W.R transition from the
coaxial connector to the stripline medium, the feed terminals
must possess an impedance of 50 Q .
The situation as the feed
terminals enter the cavity via circular holes is shown in
Fig. 2.1.7.
Inside the cavity, the feed terminal of diameter
'd' is placed at a distance b/2 from the side wall.
The
characteristic impedance of the terminal is then given by
0.213"
-- )
Fig. 2.1.5
Dimensions of a typical bandstop resonator
Fig. 2.1.6
The method of suspension of the circuit board
in the S.S.S housing
Fig. 2.1.7
The method of launching onto the S.S.S circuit
board
19
the following expression
Z
=
Log n [1.17 b/d]
(2.1.9)
/ er
For Z = 50 ft with b = 0.1 5” and er = 1
then
d
=
0.07 6”
The characteristic impedance of the transmission line
formed by the centre feed conductor and the hole in the wall
of the housing is given by
Z
=
Logio Cc/dl]
(2.1.10)
The diameter of the dielectric plug on the S.M.A connector
was 0.162".
Thus with c = 0.162",
er = 1 and Zo = 50ft we
obtain
dl
2.1.3
=
0.07"
Measured Performances of the Bandstop Resonators
The resonant frequency, 3 dB
bandwidth and stopband
insertion loss of five bandstop resonators were measured using
a Hewlett Packard swept amplitude analyser system.
measured performances are shown in Fig. 2.1.8.
showed that deep notches with 3 dB
The
The results
bandwidths of a little as
0.25% of the centre frequency are possible.
All the resonators
Fig.
2.X8
Notch
3dB Bandwidth
Inserticn loss
MHz
dB's
x lO-3
Resonant
Frequency
GHz
4.1
4.7
51
23
10
4.81
30
17.5
14.7
4.89
23
15.5
20.6
4.93
15
12.5
26
4.98
13
11
Coupling gap
Measured performance of untuned bandstop
resonators
Fig.
2.1.9
Notch
3dB Bandwidth
Inserticn Loss
MHz
dB's
xlO-3
Resonant
Frequency
GHz
4.1
4
37
19
10
4
22
17
14.7
4
10
14
20.6
4
11
10.5
26
4
8.5
Coupling Gap
8
Measured performance of bandstop resonators
after tuning to 4 GHz
20
exhibited a passband return loss of at least 15 dB
to 7GHz.
from 2GHz
Attempts to tune the resonators using capacitive
tuning screws located through the housing proved successful
but tuning down to 4GHz reduced the 3dB bandwidth of the
resonators by approximately 35%.
Measured results for all
five resonators after tuning to 4GHz are shown in Fig.
2.1.4
2.1.9.
The Equivalent Circuit of the Bandstop Resonator
By using the results presented in Fig. 2.1.8 it is
possible to evaluate the relationship between the resonator
coupling gap 'u' and its associated lumped capacitance.
First,
it is necessary to obtain the reference plane locations
of the resonator in order to accurately determine its length.
The reference plane location in the region of the coupling
gap was assumed to be similar to that of a stripline T junction
[2.4].
That is, the resonator was assumed to extend into the
main line to a distance of one quarter of the width of the
main line [Fig.
resonator,
2.1.10].
At the open circuited end of the
fringing capacitance increases the effective length
of the resonator.
To a good approximation, this situation can
be treated as a two dimensional problem with the fringing at
the end of the resonator being equivalent to that from the
edges of an isolated stripline bar [Fig. 2.1.11].
From Fig.
2 .1.11 we obtain the total capacitance per unit width of the
resonator, which is
c
£
2Cf '
£
4L'
b
(2 .1 .11 )
T
Wj
4
Fig. 2.1.10
Reference plane location of the bandstop
resonator in the region of the coupling gap
Fig.
2.1.11
Fringing capacitance off the end of a resonator
21
Where 1' is the effective length of the resonator including
fringing effects.
L'
Thus
=
L
+
bCf 1
=7^
2£
(2.1.12)
For thin strips we have
Cf 1
—
=
0.46
L'
=
L + 0.23b
G
And
(2.1.13)
Thus the total effective length, including both end effects
of the resonator is
L'
=
L + 0.23 b
+
+ U
(2.1.14)
By analysing the equivalent circuit of the resonator
with the modified electrical length it was then possible to
deduce the capacitance of the various coupling gaps.
This
was achieved simply by varying the coupling capacitance C
until the computed 3dB bandwidth of the resonator agreed with
a measured value, the value of C then corresponded to the
capacitance of the coupling gap.
The relationship between
C and the coupling gap, normalised to 1" wide resonators and
a 1" ground plane spacing is shown in Fig.
2.1.12.
u/b
Fig. 2.1.12
The relationship between the dimensions of
the resonator coupling gap and its associated
lumped capacitance.
The capacitance is
normalised to 1" wide resonators
22
2.2
Varactor Tuned Bandstop Filters
2.2.1
Choice
of Varactor Diode
As discussed in chapter 1 it was necessary to use
GaAs varactors to achieve the highest possible Q factors.
It was also necessary to use the lowest possible capacitance
values available in order to obtain realisable resonator
element values and also to avoid the possibility of the
varactors series resonating with their parasitic package
inductance.
Although varactor diode chips are available with 8:1
capacitance ratios, this value will be considerably reduced
when the varactor is enclosed in a suitable microwave package.
For example, a typical microwave package will have approxi­
mately 0.2pF capacitance, and this will appear in parallel
with the varactor chip capacitance.
Thus a varactor with a
zero bias capacitance of lpF and an 8:1 capacitance ratio
will only exhibit an external capacitance ratio of 4:1 when
enclosed in such a package.
Initially the DC4373B varactor, manufactured by AEI
Semiconductors Ltd. was chosen.
This device has a zero bias
capacitance of 0.8pF, a chip capacitance ratio of 6.5:1 and
a claimed zero bias Q factor of 6 at lOGKz.
This varactor is
enclosed in a LID package which is eminently suitable for
S.S.S circuits because of its small size [Fig. 2.2.1], in
addition the capacitance of this package is only O.llpF.
0.017"
0.08"
0.015"
<---- »
0.034"
Fig.
2.2.1
The LID varactor package
Rj (v)
Fig.
2.2.2
Fig. 2.2.3
Cj (v)
Equivalent circuit of packaged varactor
Equivalent circuit model of a shunt
connected varactor at its series resonant
frequency
23
2.2.2
Measurements of Varactor Diodes
Measurements of the DC4373B varactor at 1MHz using
a capacitance bridge confirmed the manufacturers claims with
regard to capacitance ratio and zero bias capacitance.
Dyer [2.5] has shown experimentally that 1 MHz
capacitance
measurements of varactors are accurate up to 18GHz.
In order to verify the manufacturers claims with
regard to the varactor Q factor it was necessary to measure
the varactor's small signal zero bias resistance.
this the diode was forward biased to 1 Volt.
To achieve
At this voltage
the depletion region width was zero and the resistance of
the diode is then the resistance of the undepleted epilayer
plus the substrate and contact resistances.
It was then
possible to derive the zero bias resistance of the varactor
by including the effect of a finite epilayer thickness.
method yielded a zero bias resistance of 4.6 ft .
This
This result
contradicted the manufacturers claim of approximately 3 ft.
However the method of forward biasing the varactor produced
a large dc current through the diode which would heat it, thus
producing an increase in the epilayer resistance.
To achieve more accurate measurements of the varactor
it was necessary to perform measurements at microwave frequen­
cies.
in Fig.
The equivalent circuit of the packaged device is shown
2.2.2 where Lp is the inductance of the bondwire
connecting the chip to the package and Cp is the package
capacitance.
If the diode were shunt connected to ground from
a transmission line, then at its series resonant frequency it
24
would appear as in Fig. 2.2.3.
The package capacitance will
possess a much higher impedance than the diode resistance at
microwave frequencies and thus the diode can be approximated
as a shunt resistor.
The resistance can then be obtained
by insertion loss measurements at the resonant frequency.
The insertion loss for a shunt resistor in a 50 ft system can
be expressed by
LA
=
20 Log
_LU
[1 + - ^ - 1
K.
(2.2.1)
A diode test unit was constructed in S.S.S for the purpose
of this measurement.
The circuit layout shown in Fig. 2.2.4
consisted of a 50 ft line with the diode connected from the
line to a grounded conductor.
Grounding was achieved by clamp­
ing the conductor between the side walls of the S.S.S housing.
In addition a four section stepped impedance transmission
line filter was used to provide R.f isolation between the
varactor and the D.C power supply.
The design of bias filters
is discussed in more detail later in this chapter.
The varactor resistance was obtained from the insertion
loss measurements and was 2.9ft
at zero bias and 2.5ft
at
minus 20 volts bias.
The series resonant frequency at which the insertion
loss measurement was performed was 5.9 GHz at zero bias.
The
value of the bondwire inductance was obtained from
Lp
Thus
=
Lp - 900 pH
Cj (o)
(2.2.2)
grounded strip
X
Fig.
2.2.4
D,C power supply
Test unit for measuring the equivalent circuit
parameters of a varactor
25
The measured parameters of the varactor circuit model
are used extensively in later chapters for the purpose of
computer analysis of varactor tuned filters.
2.2.3
Varactor Tuned Bandstop Resonators
By terminating the bandstop resonator discussed in
section 2.1 in a varactor diode it is possible to tune the
resonator.
In order to isolate the varactor from the dc
power supply a bias filter or choke must be used.
A three
section quarter wave stepped impedance transmission line
filter was constructed in S.S.S in order to ensure that the
bias filter would be adequate.
have a quarter wave
The filter was designed to
(stopband centre)
frequency of 6.5 GHz
and the line impedances were 15 ft, 150 ft and 15 ft.
The low
impedance 15 ft impedance lines required considerable capacitance
to ground, thus a low ground plane spacing was necessary,
fact a 0.04" ground plane spacing was used.
in
This would not
produce very high Q transmission lines but since the filter
was a broadband device this was not considered a problem.
The transfer matrix of this filter normalised to a 1 ft
system is
cos(e)
jO.3sin(0)
j3.33sin(0)
cos(0)
caste)
j 3 sin(9)
j0.33sin(6) cos (6)
cos (0)
jO.3sinC0)
j3.33sin(0) cos(0)
(2.2.3)
26
At the stopband centre frequency 6 = 90
and the
matrix reduces to
0
j0.3
0
j3
0
jO.3
(2.2.4)
j3.33
0
jO.33
0
j3.33
0
p
0
-jO.33
(2.2.5)
—j33.3
0
the insertion loss formula of a network
A ,B ,C ,D parameters we have
LA
=
10 Log
=
25 dB
[1 + 0.25 (.33.3 - 0.33)
]
(2 .2 .6 )
That is the midstopband insertion loss of this filter
will be 25 dB.
The actual measured performance of this
filter, shown in Fig.
2.2.5 exhibited greater than 20 dB loss
from 4GHz to 8GHz, the maximum rejection was not 2 5 dB
but
23 d B , the discrepancy probably being caused by the discontin­
uities at the junctions between the quarter wave sections of
transmission line.
This value was however sufficient for a
varactor tuned filter since less than 1% of the R.f power
incident upon the varactor could escape into the bias supply.
Fig.
2.2.5
Frequency
response
of
H
stepped
*1
r
Ne
cn
impedance
9
Ln
(J1
N)
o
(SJ
U1
G 1.E
oT
transmission
line
filter
27
Two varactor tuned bandstop resonators were constructed
using the resonators described previously and the bias filters
described above.
The coupling gaps between the resonators
and the main line were 0.005" and 0.008" corresponding to
percentage 3 dB
bandwidths of 06% and 1% at 4GHz .
The bias
filters were 2 section quarter wave filters with the low
impedance lines being realised by a sandwich structure of
dielectric circuit board and conductor clamped between the
side walls of the
S.S.S housing [Fig. 2.2.6]
The measured performances of these resonators are shown
as a function of varactor bias voltage in Fig. 2.2.7.
These
results show that a tuning bandwidth of approximately 25% of
the upper tuned frequency is possible.
Unfortunately the Q
factor of these resonators was insufficient to produce an
acceptable response, the worst case stopband insertion loss
was only 1 .4 d B .
There are two methods to overcome the problem of low
resonator Q factor, the most obvious is to increase the band­
width of the resonator.
Of more interest is the method of
decoupling the varactor from the resonator by connecting it
in series with a lossless capacitor.
1 pF
For example, consider a
varactor with a 4:1 capacitance ratio connected in series
with a 1 pF
lossless capacitor.
terminal capacitor is
0.5 pF
At zero bias the effective
and at maximum bias it is 0.2 pF .
Thus the Q factor at zero bias has been doubled albeit the
capacitance ratio has been reduced from 4:1 to 2.5:1.
technique would therefore seem to be of some value.
This
conductor
F i g . 2.2.6
dielectric
Realisation of bias choke with a sandwich structure
Bias
-V
0
Stopband
insertion
1.9
loss
dB's
2
3
4
5
10
15
20
30
2.5 3.5
4
4.5
4.9
6.5
7
7.5
8.9
1
Centre
frecruency 3.04 3.34 3.45 3.53 3.64 3.68 3.78 3 & 5 3.89 3.95
GHz
3 dB
stopband
bandwidth
MHz
a)
Bias
-V
74
Coupling gap
0
stopband
insertion
1.4
loss
dB's
1
1.8
2
=
74
75
77
10
15
20
30
5.5
6.5
0.005"
4
3
74
2.1 2.8 3,3
5
3.5 4.5 5.2
centre
frequency
3.06 3.36 3.5 3.59 3.66 3.72 3.83 3.91 3.97 3.99
GHz
3 dB
stopband
Dandwidth
MHz
b)
Fig. 2 . 2 . 1
42
Coupling gap
=
43
44
46
0.008"
Measured performances of varactor tuned bandstop
resonators
28
Several bandstop resonators have been constructed using
this technique.
The resonators were again 1.1" long and the
coupling gaps varied between 0.003" and 0.025".
Two values
of series decoupling capacitors were used, these were approx­
imately 1 pF
and 0.5 pF
and were realised by overlapping
conductors on opposite sides of the S.S.S circuit board.
The
overlapping capacitors were designed as if they were simple
parallel plate capacitors with a fringing capacitance of 30%
of the parallel plate capacitance.
The measured performances of these varactor tuned
resonators are shown in Fig.
2.2.8.
The results show that
with the varactors decoupled via the 1 pF
capacitor, tuning
bandwidths of approximately 12% are achieved with acceptable
responses for resonator 3 dB
bandwidths of greater than 2%.
With the lower value of decoupling capacitor tuning bandwidths
of 7% are possible and insertion loss levels of greater than
7 dB
are maintained for resonators with 3 dB
greater than 1%.
bandwidths of
In conclusion it can be seen that useful
varactor tuned resonators with low dissipation loss are achieved
with tuning bandwidths of approximately seven times their 3 dB
bandwidth.
2.2.4
Varactor Tuned Multi-Resonator Bandstop Filters
Several two section bandstop varactor tuned filters
were constructed in order to investigate the effects of tuning
on filter performance.
The filters were constructed using the
resonators discussed in section 2.2.3 which were separated by
Bias
OV
3.54
36
4.5
6.5
-30V
3.88
45
11.5
2.4
(i)
Bias
Strong varactor coupling
Stopband Stopband
Centre
3 dB
insertion return
frequency bandwidtt
loss
loss
GHz
MHz
dB
dB
0V
3.65
46
9
-30 V
3.82
49
13
(ii)
Fig.
Stopband Stopband
Centre
3dB
frequency bandwidth insertior return
loss
loss
GHz
MHz
dB
dB
2.2.8 b)
3.8
2
Weak varactor coupling
Measured performance of a varactor tuned
resonator with a coupling gap of 0 .0 1 "
Bias
OV
3.5
30
4.5
-30 V
3.84
40
10,5
(i)
Bias
8.5
3
Strong varactor coupling
Stopband Stopband
3 dB
Centre
insertion
return
frequency bandwidth
loss
loss
MHz
GHz
dB
dB
OV
3.55
34
8
-30 V
3.73
40
13
(ii)
Fig. 2.2.8c
Stopbanc Stopband
Centre
3 dB
inserticn return
frequency bandwidtl
loss
loss
MHz
GHz
dB
dB
4.5
2
Weak varactor coupling
Measured performance of a varactor tuned
resonator with a coupling gap of 0.018"
stonband stopband
Centre
3dB
return
insertior
frequency bandwidth
loss
loss
GHz
MHz
dB
dB
Bias
OV
3.53
15
3.5
14
-30V
3.94
30
7.5
4.5
(i)
Bias
Strong varactor coupling
stopband stopband
3 dB
Centre
insertion return
frequency bandwidth
loss
loss
MHz
GHz
dB
dB
OV
3.81
25
5
7
-30V
3.93
30
10
4
(ii)
Fig. 2.2.8 d
Weak varactor coupling
Measured performance of a varactor tuned
resonator with a coupling gap of 0.02 5"
29
a 50 ft t r a n s m i s s i o n line w i t h 90° p h a s e l e n g t h at 3.3GHz.
T h e r e f e r e n c e p l a n e s of the t r a n s m i s s i o n line w e r e a s s u m e d
to be at t he e x a c t c e n t r e of the r e s o n a t o r s as s h o w n in Fig.
2.2.9.
F igs.
Th e m e a s u r e d p e r f o r m a n c e s of six f i l t e r s are s h o w n in
2 . 2 . 1 0 - 2.2.15, t h e s e i n c l u d e d e s i g n s for t h r e e d i f f e r e n t
b a n d w i d t h s and tw o d i f f e r e n t v a l u e s of v a r a c t o r d e c o u p l i n g
capacitor.
The m e a s u r e d r e s u l t s s h o w t h a t the f i l t e r s e x h i b i t
an i n c r e a s e in s e l e c t i v i t y o v e r t h a t of the s i n g l e r e s o n a t o r s ,
a n d an i n c r e a s e in s t o p b a n d i n s e r t i o n loss.
H o w e v e r , as in
the c a s e of the s i n g l e r e s o n a t o r s , g o o d r e s p o n s e s w e r e onl y
a c h i e v e d f or t u n i n g b a n d w i d t h s of 10% or less.
In e v e r y c a s e
t h e f i l t e r s c o u l d be t u n e d w i t h a s i n g l e p o w e r s u p p l y p r o v i d e d
that some initial tuning using tuning screws was performed.
It c a n be s e e n t h a t the f r e q u e n c y r e s p o n s e o f e a c h f i l t e r
varies
as a f u n c t i o n of t u n e d f r e q u e n c y .
T h i s is p a r t l y d u e
to the c h a n g e in v a r a c t o r Q f a c t o r w i t h b i a s v o l t a g e a n d a l s o
d u e to the c h a n g e in p h a s e s h i f t b e t w e e n the r e s o n a t o r s .
In o r d e r to c o n s t r u c t h i g h p e r f o r m a n c e b a n d s t o p f i l t e r s
it is n e c e s s a r y to h a v e a c c u r a t e k n o w l e d g e of the l o c a t i o n of
the r e f e r e n c e p l a n e s of the m a i n t r a n s m i s s i o n line b e t w e e n the
resonators.
These were measured using a Hewlett-Packard network
a n a l y s e r a n d ar e s h o w n in Fig. 2.2. 1 6 .
plane locations a
4 resonator bandstop filter was designed
f o r o p e r a t i o n at 10GHz.
were
Using these reference
The p h a s e l e n g t h s b e t w e e n t h e r e s o n a t o r s
e x a c t l y 90° at 1 0 G H z .
T h e d e s i g n of the r e s o n a t o r s w a s
s o m e w h a t i n t u i t i v e , t h e s e b e i n g all the same l e n g t h and the
c o u p l i n g g a p s b e t w e e n the o u t e r r e s o n a t o r s a n d the m a i n line
w e r e 2.5 t i m e s l a r g e r t h a n the i n n e r r e s o n a t o r c o u p l i n g gaps.
Reference plane
w
Fig.
2.2
f-
Th e a s s u m e d r e f e r e n c e p l a n e l o c a t i o n s of the
r e s o n a t o r s for the v a r a c t o r t u n e d b a n d s t o p
filters
varactor
tuned
filter
response
with
a coupling
gap
>
2 Stage
LD
CN
2.2.10
of
0.005
U)
rd
Fig.
lD
CN
O
I
Fig.
2.2.11
2 Stage
coupled
varactor tuned
varactors
filter
response
with
a coupling
gap
of
0.005"
and
weakly
Fig.
2.2.12
2 stage
varactor
31.56 Glz
tuned
filter
response
with
0.01"
-200 MHz
gaps
_ ._ __
3.47 GHz
coupling
-100 MHz
+100 MHz
+200 MHz
1
-100 MHz
LO
I
r0—
I
1
Lfl
stage
varactor
tuned
filter
with
2
s ss«
CO
0.018"
S
response
coupling
gaps
8 5?n
CM
N
CN
CN
tn
-H
Cm
a safe
0)
14
2 stage
varactor
tuned
filter
response
with
0.018"
coupling
gaps
a ss'e
Reference
planes
w/10
—-Ma—
1 r
!i V
1M —
‘
W^IO
III
I
e w/M
Fig.
2.2.16
M e a s u r e d r e f e r e n c e p l a n e l o c a t i o n s of the
bandstop resonators
30
The S.S.S circuit
Fig.
2.2.17.
b o a r d l a y o u t o f this f i l t e r is s h o w n in
Th e f i l t e r w a s m e c h a n i c a l l y t u n e d to a c e n t r e
f r e q u e n c y of 1 0GHz a n d the m e a s u r e d p e r f o r m a n c e is s h o w n in
Fig.
2.2.18.
The
f r e q u e n c y r e s p o n s e of the f i l t e r e x h i b i t e d a
g r e a t e r s e l e c t i v i t y and a l ower l e v e l of p a s s b a n d r e t u r n loss
o n th e h i g h e r s i d e of the s t o p b a n d t h a n on t h e l o w e r side.
A p o s s i b l e e x p l a n a t i o n of this w o u l d b e t h a t the p h a s e s h i f t
b e t w e e n the r e s o n a t o r s w e r e i n c o r r e c t .
c o u p l e d by a t r a n s m i s s i o n line
transfer matrix
w i t h the
j sin (0)
=
(2.2.7)
co s (0)
jsin(6)
Where
(assumed l o s s l e s s )
( n o r m a l i s e d to a 1 ft system) b e l o w .
cos (.6)
[T]
The resonators were
0
=
aw
T h i s m a t r i x c an be d e c o m p o s e d i n t o the p r o d u c t b e l o w ,
/sin(0)
0
/sin (0)
1
0
0
0
-jcos (0)
1/ /sin (0)
0
0
( 2. 2 . 8)
-jcos(0)
1
0
/sin (0)
r
2.2.17
i
Fig.
Dimensions
of
4 stage
bandstop
f i lter
o
O
CN
cn
a
1s a
Fig.
2.2.18
Measured
performance
of
four
stage
bandstop
f i lter
31
T h e m a t r i x p r o d u c t of 2.2.8 r e p r e s e n t s an i d e a l u n i t y
i m p e d a n c e i n v e r t e r s h u n t e d o n each s i d e b y f r e q u e n c y d e p e n d e n t
r e a c t a n c e s an d t e r m i n a t e d in f r e q u e n c y d e p e n d e n t t r a n s f o r m e r s
[Fig. 2 . 2 . 1 9 ] .
If the t r a n s m i s s i o n line w e r e e x a c t l y 90°
long t h e n 2.2.8 w o u l d d e g e n e r a t e into the r e q u i r e d u n i t y
impedance inverter.
F o r l e n g t h s of line o t h e r t h a n 90° the
s h u n t r e a c t a n c e s b e c o m e f i n i t e and load the r e s o n a t o r s , thus
c a u s i n g the a s y m m e t r i c f r e q u e n c y r e s p o n s e , a l s o the t r a n s f o r m e r s
w i l l no l o n g e r h a v e u n i t y turns r a t i o and the f i l t e r p a s s b a n d
r e t u r n loss w i l l be d e g r a d e d .
T h i s e f f e c t is s h o w n in Fig.
2 . 2 . 2 0 fo r a p r o t o t y p e b a n d s t o p f i l t e r
w i t h the c o m p u t e d
f r e q u e n c y r e s p o n s e g i v e n for p h a s e s h i f t s b e t w e e n r e s o n a t o r s
of 60°, 90° an d 120°.
However,
the f i l t e r r e s p o n s e s h o w n in Fig. 2 . 2 . 1 9 was
e v i d e n t w h e n the p h a s e s h i f t b e t w e e n the r e s o n a t o r s w a s
e x a c t l y 90°.
The c o r r e c t e x p l a n a t i o n of t h i s p h e n o m e n o n can
be a t t r i b u t e d to th e b a s i c p r o p e r t i e s of the r e s o n a t o r s .
The
e q u i v a l e n t c i r c u i t of the b a n d s t o p r e s o n a t o r i n c l u d i n g a
l u m p e d t u n i n g c a p a c i t o r is s h o w n in Fig.
2.2.21.
The input
i m p e d a n c e of this r e s o n a t o r is
Zintjw) =
j[Zo2C 1C2w2tantaw) -wCCl+C2)Zo -tan(aw) ]
^
2 9)
wCl[tan(aw) + wC2Zo]
The resonator possesses a transmission zero and a
t r a n s m i s s i o n p o l e , t h e s e b e i n g d e f i n e d by the z e r o s of the
n u m e r a t o r and d e n o m i n a t o r of 2,2,9.
The f r e q u e n c y of p e r f e c t
t r a n s m i s s i o n is s l i g h t l y g r e a t e r than the t r a n s m i s s i o n ze ro
o-----
---- °
U.E
1 ft
0-----
-----0
/sin (6); 1
Fig.
2.2.19
1: /sin (.6)
E q u i v a l e n t c i r c u i t of a u n i t e l e m e n t
of t r a n s m i s s i o n line
Fig.
2.2.20
Th e e f f e c t on b a n d s t o p f i l t e r p e r f o r m a n c e of
v a r y i n g the p h a s e s h i f t b e t w e e n the r e s o n a t o r s
Fig.
2.2.21
E q u i v a l e n t c i r c u i t of a t u n a b l e b a n d s t o p r e s o n a t o r
32
f r e q u e n c y , thus the i m p e d a n c e of the r e s o n a t o r w i l l c h a n g e
m o r e r a p i d l y for f r e q u e n c i e s a b o v e the t r a n s m i s s i o n zero
f r e q u e n c y t h a n b e l o w it.
frequency response
T h i s has the e f f e c t of m a k i n g the
of a b a s i c r e s o n a t o r c o u p l e d to a m a i n
line m o r e s e l e c t i v e o n the h i g h side of the r e s o n a n t f r e q u e n c y
t h a n b e l o w it.
assymmetric,
T h u s s i n c e the r e s o n a t o r s are i n h e r a n t l y
the f r e q u e n c y r e s p o n s e of a m u l t i s t a g e b a n d s t o p
f i l t e r w i t h r e s o n a t o r s c o u p l e d by 90° p h a s e s h i f t e r s w i l l
a l s o be a s s y m e t r i c .
In c h a p t e r three, a t e c h n i q u e is c o n s t r u c t e d for
e v a l u a t i n g th e c o r r e c t p h a s e s h i f t b e t w e e n the b a n d s t o p
r e s o n a t o r s to c o m p e n s a t e for t heir i n h e r e n t a s s y m e t r y .
2.3
Tunable Microwave Bandpass Filters
There are many bandpass filters suitable for realisat­
io n in S.S.S.
T h e s e include, for exam p l e ,
the c o m b l i n e f i l t e r
[2.6], the i n t e r d i g i t a l f i l t e r [2.7] and the c a p a c i t i v e g a p
c o u p l e d t r a n s m i s s i o n line f i l t e r [2.8].
An initial decision
w a s m a d e to a v o i d s h o rt c i r c u i t e d t r a n s m i s s i o n l i n e s in the
S . S . S m e d i u m as t h e s e w o u l d c o n t r i b u t e to the f i l t e r loss.
T h e o t h e r m a j o r c o n s i d e r a t i o n w a s that v a r a c t o r d i o d e s and
t h e i r b i a s f i l t e r s m u s t be e a s i l y i n c o r p o r a t e d i n t o the c i r c u i t .
W i t h the a b o v e c o n s i d e r a t i o n s in m i n d the i n i t i a l c h o i c e
of f i l t e r w a s the S . S . S s t r u c t u r e s h o w n in Fig.
2.3.1.
Basically,
this f i l t e r is c o m p o s e d of o p e n c i r c u i t e d t r a n s m i s s i o n line
r e s o n a t o r s of b e t w e e n one q u a r t e r and o n e h a l f w a v e l e n g t h long
c a p a c i t i v e l y c o u p l e d to u n i t e l e m e n t s of t r a n s m i s s i o n line. The
capacitive
coupling gap
coupling unit
element
open circuited
stub
Fig.
2.3.1
S t r u c t u r e of S . S . S b a n d p a s s f i l t e r
1
0.25"
T
0.005"
0.9"
1 . 2"
J
0.25"
Fig.
2.3.2
D i m e n s i o n s of two s t a g e b a n d p a s s f i l t e r
33
e q u i v a l e n t c i r c u i t of this f i l t e r is n o t p r e s e n t e d s ince
e x a c t d e s i g n t e c h n i q u e s are n o t d i s c u s s e d in this chap t e r .
A tw o r e s o n a t o r b a n d p a s s f i l t e r w a s c o n s t r u c t e d for
the p u r p o s e of i n v e s t i g a t i n g the e f f e c t s of tuning.
S . S.S
2.3.2.
The
c i r c u i t b o a r d l a y o u t of this f i l t e r is s h o w n in Fig.
T h e r e s o n a t o r s w e r e d e s i g n e d to be 13 5° long a t p a s s ­
b a n d c e n t r e and the u n i t e l e m e n t c o u p l i n g the r e s o n a t o r s w a s
7 5° long.
T h e c a p a c i t i v e gaps c o u p l i n g the u n i t e l e m e n t to
the r e s o n a t o r s w e r e m a d e 0.01" w i d e as this w a s r o u g h l y the
s a m e v a l u e as t h a t u s e d for d e c o u p l i n g b a n d s t o p r e s o n a t o r s
in s e c t i o n 2.2.
T h e m e a s u r e d p e r f o r m a n c e of this f i l t e r is
s h o w n for t wo v a l u e s of t u n e d f r e q u e n c y in Fig.
2.3.3. T u n i n g
w a s a c h i e v e d b y c a p a c i t i v e t u n i n g s c r e w s l o c a t e d o v e r the
ends of the r e s o n a t o r s .
W h e n t u n e d to 3.22 G H z t h e f i l t e r e x h i b i t e d an a c c e p t ­
a b l e f r e q u e n c y r e s p o n s e w i t h a p a s s b a n d r e t u r n loss o f 12.5 dB
a n d a b a n d w i d t h of 43 M H z .
1.8 dB
T h e p a s s b a n d i n s e r t i o n loss w a s
w h i c h is c o m p a r a b l e to c o a x i a l f i l t e r s of t h e same
p h y s i c a l size.
[The g r o u n d p l a n e s p a c i n g w a s 0 . 3 " ] .
w h e n t h e f i l t e r w a s t u n e d to 2.8 G H z
However
the r e s p o n s e d e t e r i o r a t e d
d r a s t i c a l l y w i t h the d i s s i p a t i o n loss b e c o m i n g v e r y high.
T h i s p h e n o m e n o n is k n o w n as u n d e r c o u p l i n g
[2.9] and is a
r e s u l t of the c o u p l i n g b e t w e e n the r e s o n a t o r s b e i n g h i g h l y
d e p e n d e n t on f r e q u e n c y .
A t e c h n i q u e for o v e r c o m i n g this
p r o b l e m is d e v e l o p e d in c h a p t e r four.
An attempt was m a d e
b a n d p a s s filter.
to varactor tune the two resonator
D C 4 3 7 3 B v a r a c t o r s w e r e u s e d and t h e y w e r e
d e c o u p l e d b y 0 . 5 p F c a p a c i t o r s to i n c r e a s e t h e i r Q factor.
cn
9 e
Measured
a)
Fig.
2.3,3
-50 MHz
-100 ffiz
performance
3.22 GHz
of
2 stage
bandpass
+50 MHz
filter
+100 MHz
b)
sI
8
in
frequency
of
2 stage
CO
performance
js
—
Measured
O
2.3.3
bandpass
filter
s
Fig.
after
tuning
+
CN
down
Fig. 2.3.4
of
cn
5
1
S 1
tuned
bandpass
filter
+kMiz
at
zero
bias
+100 MHz
O
ro
varactor
3.1GHz
m
<n
performance
-50 MHz
O
<N
Measured
-100 MHz
Ln
,H
34
T he m e a s u r e d p e r f o r m a n c e of this f i l t e r at zero b i a s is
s h o w n in Fig.
2.3.4.
A p a s s b a n d i n s e r t i o n loss of 2 . 5dB a n d
a t u n i n g b a n d w i d t h of 8% w e r e a c h i e v e d .
The f r e q u e n c y
r e s p o n s e of this f i l t e r d e t e r i o r a t e d r a p i d l y w i t h t u n i n g as
e x p e c t e d b u t the loss p e r f o r m a n c e at z e r o b i a s was e n c o u r a g ­
ing an d s u g g e s t e d t h a t w i t h a s u i t a b l e f i l t e r d e s i g n , u s e f u l
low loss v a r a c t o r t u n e d b a n d p a s s f i l t e r s w e r e f e a s i b l e .
2.4
Conclusions
2.4.1
V a r actor Loss
E x p e r i m e n t a l r e s u l t s s h o w that l o w loss v a r a c t o r
t u n e d tw o s e c t i o n b a n d s t o p a n d b a n d p a s s f i l t e r s c a n be
constructed,
and t h e y w i l l e x h i b i t t u n i n g b a n d w i d t h s of
a p p r o x i m a t e l y 10%.
T o i m p r o v e u p o n t h e s e r e s u l t s it is
n e c e s s a r y to u s e h i g h e r q u a l i t y v a r a c t o r diodes.
All preced­
ing w o r k in this t h e s i s i n v o l v e s the u s e of the M i c r o w a v e
Associates MA46620G varactor.
This diode has a similar
c a p a c i t a n c e a n d c a p a c i t a n c e v a r i a t i o n to t h e A E I 4 3 7 3 B
v a r a c t o r bu t h a s an e p i t a x i a l l a y e r r e s i s t a n c e of o n l y 1.25 ft
at z e r o bias.
T h i s c o r r e s p o n d s to a 2.4 t i m e s i n c r e a s e in Q
f a c t o r as c o m p a r e d w i t h the A E I v a r a c t o r .
In a d d i t i o n al l the
f i l t e r s d e s c r i b e d in the r e m a i n d e r
of this t h e s i s w e r e d e s i g n e d for 3dB b a n d w i d t h s of at l east
5%.
C o m b i n e d w i t h the h i g h e r q u a l i t y v a r a c t o r s t h i s e n a b l e s
m u c h b r o a d e r t u n i n g b a n d w i d t h s to be a c h i e v e d .
35
2.4.2
Circuit Design Techniques
A l l t h e f i l t e r s d e s c r i b e d in this c h a p t e r w e r e
d e s i g n e d on a l a r g e l y i n t u i t i v e b a s i s w i t h the o b j e c t i v e
m e r e l y b e i n g to o b s e r v e the e f f e c t s of v a r a c t o r loss and
tuning on filter performance.
Experimental results show
t h a t a t e c h n i q u e for e v a l u a t i n g the c o r r e c t p h a s e s h i f t
b e t w e e n the r e s o n a t o r s o f b a n d s t o p f i l t e r s is r e q u i r e d and
t h a t a b a n d p a s s f i l t e r c i r c u i t s u i t a b l e for b r o a d b a n d t u n i n g
must
be d e v e l o p e d .
In a d d i t i o n s i n c e loss is a m a j o r c a u s e
of th e d e t e r i o r a t i o n of v a r a c t o r t u n e d f i l t e r p e r f o r m a n c e ,
t e c h n i q u e s of p r e d i c t i n g its e f f e c t m u s t be d e v e l o p e d .
References
2.1
' C o u p l e d r e c t a n g u l a r b a r s b e t w e e n p a r a l l e l plat e s '
W.J. G e t s i n g e r
IRE. T R A N S M . I . T . J A N 1962
2.2
'Microwave filters,
i m p e d a n c e m a t c h i n g n e t w o r k s and
coupling structures', ppl68-172
Matthaei, Young and
Jones, M c G r a w H i l l
2.3
R e f 2.2
pp243-244
2.4
J.E. D e an, Ph.D. T h e s i s
D e pt, of E l e c t r i c a l a n d E l e c t r o n i c E n g i n e e r i n g
T h e U n i v e r s i t y of L e e d s
2 .5
G.R. D yer, Ph.D. T h e s i s
D e p a r t m e n t of E l e c t r i c a l a n d E l e c t r o n i c E n g i n e e r i n g
T h e U n i v e r s i t y of L e e d s
36
2.6
Re f 2.2
p p 4 9 7 - 5 05
2.7
Re f 2.2
p p 61 4 - 6 25
2.8
R e f 2.2
pp440-450
2.9
Ref 2.2
pp663-668
A p p e n d i x 2.1
T h e g r a p h s o n the f o l l o w i n g p a g e s s h o w the e v e n and
o d d m o d e f r i n g i n g c a p a c i t a n c e s of c o u p l e d r e c t a n g u l a r b a r s
b e t w e e n p a r a l l e l g r o u n d p l a n e s as a f u n c t i o n of b a r s e p a r a t ­
ion.
T h e y a l s o s h o w th e c o u p l i n g c a p a c i t a n c e b e t w e e n t h e
ba r s a n d the f r i n g i n g c a p a c i t a n c e f r o m an i s o l a t e d bar.
37
0.2
03
0 4
0.5
06
0 7
•1-0
0 01
•»<3 * OV
Even mode fringing capacitance and coupling capacitance
between coupled rectangular bars
UNI'” SSITY LIBRARY LEEDS!
38
F r i n g i n g c a p a c i t a n c e f r o m an i s o l a t e d r e c t a n g u l a r
bar
0 2
0 3
0 4
0 5
0 6
0 7
0 8
0 9
39
001
O d d m o d e f r i n g i n g c a p a c i t a n c e of c o u p l e d r e c t a n g u l a r
bars
40
3
TUNABLE MICROWAVE BANDSTOP FILTERS
3.1
Introduction
In c h a p t e r two it w a s s h o w n t h a t in o r d e r t o a c h i e v e
the o p t i m u m s y m m e t r i c a l b a n d s t o p f i l t e r r e s p o n s e ,
it is
n e c e s s a r y to c o u p l e the r e s o n a t o r s w i t h t r a n s m i s s i o n l i nes
o f p h a s e s h i f t n o t e q u a l to 90°.
In this c h a p t e r a d e s i g n
procedure has been developed which enables the correct phase
s h i f t s b e t w e e n r e s o n a t o r s to b e e v a l u a t e d .
Two fixed frequency
S.S.S bandstop filters have been designed and constructed
u s i n g this t e c h n i q u e .
B o t h f i l t e r s w e r e d e s i g n e d f r o m an
i n v e r s e C h e b y s h e v p r o t o t y p e a n d h a d a c e n t r e f r e q u e n c y of
4 GHz
a n d r e s p e c t i v e 40 dB
s t o p b a n d b a n d w i d t h s o f 1% a n d 10%.
Both devices performed extremely well, e x h i biting the required
s y m m e t r i c a l f r e q u e n c y r e s p o n s e , b r o a d low V . S . W . R p a s s b a n d s
a n d h i g h l e v e l s of s t o p b a n d i n s e r t i o n loss.
A computer
a n a l y s i s p r o g r a m w a s d e v e l o p e d to e x a m i n e t h e e f f e c t s of
d i s s i p a t i o n loss a n d t he c o m p u t e d r e s u l t s s h o w e d e x c e l l e n t
a g r e e m e n t w i t h th e m e a s u r e d p e r f o r m a n c e s .
A varactor tuned bandstop filter has been designed
u s i n g the n e w p r o c e d u r e f o r c o m p u t i n g the p h a s e s h i f t s b e t w e e n
the filter resonators.
In this c a s e t h e o p t i m u m p h a s e s h i f t
w a s c h o s e n to b e a t t h e e x a c t c e n t r e o f the t u n i n g b a n d w i d t h .
T h e t u n a b l e f i l t e r w a s a t h r e e r e s o n a t o r d e v i c e d e s i g n e d to
t u n e f r o m 3.5 GHz
to 4.5 GHz
with a
1% 20 dB
stopband bandwidth.
A c o m p u t e r a n a l y s i s p r o g r a m h a s b e e n d e v e l o p e d in o r d e r t h a t
t h e e f f e c t s of v a r a c t o r loss c a n b e p r e d i c t e d a n d t h e c o m p u t e d
results agree closely with the measured performance of the
v a r a c t o r t u n e d filter.
3.2
T h e o r y of S y m m e t r i c a l B a n d s t o p F i l t e r s
Th e e q u i v a l e n t c i r c u i t of the S . S . S b a n d s t o p f i l t e r
is s h o w n in Fig.
3.2.1.
The i n p u t a d m i t t a n c e o f t h e rth
r e s o n a t o r o f this f i l t e r can e a s i l y b e s h o w n to b e
Y r (j cjo)
= j [uCl^C Zr a)C2r + tan(au))]]
[a)Z (Cl + C2 ) - tan (au) [ Z 2 U2C1 C2 -1]]
r
r
r
r
r r
Y r (j“) p o s s e s s a p o l e at up^ w h i c h c o r r e s p o n d s to the rth
t r a n s m i s s i o n z e r o o f the filter.
Y r (jo)) a l s o p o s e s s e s a zero
a t a f r e q u e n c y uz^, s l i g h t l y h i g h e r t h a n ojpr .
It is this z e r o
w h i c h c a u s e s Y r (jui) to v a r y m o r e r a p i d l y at f r e q u e n c i e s a b o v e
w p ^ t h a n b e l o w wp^ a n d thus the r e s o n a t o r f r e q u e n c y r e s p o n s e
is a s y m m e t r i c .
F o r f r e q u e n c i e s c lose to u p r , y r (j<u) c a n b e a p p r o x i ­
m a t e d b y the b i l i n e a r f u n c t i o n below:
j (tO “<DZ )
Y r (joi)
K
r
(w - uip ) (ojz - up )
r
r
^r
(3.2.2)
D i f f e r e n t i a t i n g 3 . 2 . 1 w i t h r e s p e c t to w a n d e v a l u a t i n g at
copr w e o b t a i n
Fig.
3.2.1
T h e e q u i v a l e n t c i r c u i t of the S . S . S b a n d s t o p
filter
42
The v a l u e s of uz
r
a n d cop are o b t a i n e d f r o m t h e z eros
cr
o f t h e n u m e r a t o r a n d d e n o m i n a t o r of 3 .2.1 as b e low.
tan(aa)Z )/uz
r
r
- Z C2
r r
(3.2.5)
ta n (aup ) [ Z 2 up 2 Cl C2 -1] = up Z [Cl +C2 ]
*r
r
r r
*r r
r
r
The e x p r e s s i o n 3.2.2 for Y^tju)
(3.2.6)
can be decomposed by
e x t r a c t i n g the r e s i d u e a t u p r to o b t a i n
Y
r
(ju)
=
jB
J r
- j / (coL - X )
J
r
r
(3.2.7)
where
B
r
Lr
X
1/K
K
r
(uz
r
- up )
^r
r
r
(3.2.8)
(3.2.9)
(3.2.10)
Y r (ju>) h a s thus b e e n a p p r o x i m a t e d b y a r e s o n a t o r c o m p o s e d of
an i n d u c t o r a n d a f r e q u e n c y i n v a r i e n t r e a c t a n c e , w h i c h is
shunted by another frequency invariant reactance.
If t h e
shunt reactance were removed from the resonator, the remaining
r e s o n a t o r w o u l d p o s s e s s a l i n e a r r e a c t a n c e as a f u n c t i o n of
43
f r e q u e n c y a n d this w o u l d p r o d u c e the r e q u i r e d s y m m e t r i c a l
filter response.
Thus it is n e c e s s a r y to c o n s t r u c t a
t e c h n i q u e to r e m o v e
f r o m e a c h reso n a t o r .
The r e s o n a t o r is n o w d e c o m p o s e d i n t o t h e n e t w o r k
s h o w n in Fig.
3.2.2.
The n o t a t i o n in Fig.
3.2.2 is as
follows:
Y'r (ja») =
(3.2.11)
B
(3.2.12)
r
=
B ’ + B"
r
r
A p p l y i n g this p r o c e d u r e to e v e r y r e s o n a t o r a n d c o u p l i n g t h e
rth a n d r + i t h r e s o n a t o r s via u n i t y i m p e d a n c e p h a s e s h i f t e r s
w i t h p h a s e s h i f t 6r r+1 w e o b t a i n the t y p i c a l c o u p l i n g netw o r k s h o w n in Fig.
1/n
3. 2. 3 w i t h the t r a n s f e r m a t r i x b e l o w
0
1
0
0
n
jB"
J r
1
1
0
n
0
r
r
n
r+1
oos(0r,r+l)
£ s3jl(6r,rfl2_
f * (er,r+li.
(3.2.13)
JB'rH
I
_
0
1/nt+i_
A f t e r m u l t i p l y i n g o u t 3.2. 1 3 t o f o r m a s i n g l e m a t r i x w e
obtain
n
r
:1
1 :n
Fig.
3.2.2
D e c o m p o s i t i o n of the b a s i c b a n d s t o p r e s o n a t o r
Fig.
3.2.3
T h e t y p i c a l c o u p l i n g n e t w o r k b e t w e e n the
r e s o n a t o r s of the b a n d s t o p f i l t e r
44
n r+l
n
r
C08(er,r+l) ' 8’
r + l sln(6r,r+l)
jn n ,
J r r+l
j sin (e
,, v
J
r,r+l)
n n ,n
r r+l
c o s (6
.. )
r,r+l
B ; ccos(6r _r+i) - B ’
r+1s“ «>r ,r+i)
nr+l
-B" sin (0
r
r,r+l
+s“ (9r,r+l) + E 'r+ lcos(er,r+l)
(3.2.14
This m a t r i x can b e e q u a t e d t o t h a t o f an i d e a l a d m i t t a n c e
i n v e r t e r u n d e r t h e c o n d i t i o n that
cos (6
r,r + l
) = B"sin(e
r
., ) = B . ' sin (6
r,r+l
r+l
(3.2.15)
r,r+r
The a d m i t t a n c e o f t h e i n v e r t e r is
K
.,
r,r+l
=
n
r
n
, - / s in ( 6
,, )
r+l
r,r+l
(3.2.16)
The i n i t i a l c o n d i t i o n s are
n^
=
1
a n d B '^
=
0
(3.2.17)
P r o v i d e d t h a t the a b o v e c o n d i t i o n s are s a t i s f i e d the
elements B
r
in e a c h r e s o n a t o r h a v e n o w b e e n a b s o r b e d i n t o the
p h a s e s h i f t e r s thus f o r m i n g i d e a l i m p e d a n c e i n v e r t e r s b e t w e e n
th e r e s o n a t o r s .
T he d e s i g n e q u a t i o n s for the s y m m e t r i c a l b a n d s t o p
45
f i l t e r can n o w be d e v e l o p e d .
The d e s i g n is b a s e d on the
i n v e r s e C h e b y s h e v b a n d s t o p p r o t o t y p e f i l t e r s h o w n in Fig.
3.2.4.
This p r o t o t y p e is d e r i v e d f r o m the i n v e r s e C h e b y s h e v
n a t u r a l h i g h p a s s p r o t o t y p e f i l t e r u s i n g the s l o p e r e a c t a n c e
p a r a m e t e r t e c h n i q u e [3.1].
The b a n d s t o p p r o t o t y p e e l e m e n t
v a l u e s are g i v e n b y the f o l l o w i n g e q u a t i o n s
2a. + cos (0 )
(3.2.18)
4wq n s i n (er )
4 n s in (6 )
too (2a- c os (0 ) )
(3.2.19)
with
(3.2.20)
a
=
to )/(a)2-a)1 )
(3.2.21)
=
(2r-l)7r/2n
(3.2.22)
LA + L R + 6
n >
(3.2.23)
20 L o g (S + / S -1)
n
=
sinh[^ sinh
(1/e ) ]
,
c 1 Q (LA/10) _ u -)s
where
a n d co2 a re th e p r e s c r i b e d s t o p b a n d e d g e f r e q u e n c i e s
L A is t h e s t o p b a n d i n s e r t i o n loss at co^ a n d co2
(3.2.24)
(3.2.25)
Fig.
3 . 2.4
Th e I n v e r s e C h e b y s h e v p r o t o t y p e b a n d s t o p f i l t e r
46
L R is the r e t u r n loss at the p a s s b a n d e d g e f r e q u e n c i e s .
S
is the r a t i o of the p a s s b a n d to s t o p b a n d b a n d w i d t h s .
The r e s o n a n t f r e q u e n c i e s a n d b a n d w i d t h s of t h e p r o t o ­
t y p e r e s o n a t o r s are g iven b y
0)0
(3.2.26)
/l T
r r
Aw
(3.2.27)
2L
In o r d e r t o o b t a i n the e l e m e n t v a l u e s f o r t h e m i c r o ­
w a v e r e s o n a t o r s , t h e e x p r e s s i o n s for t h e i r c e n t r a l f r e q u e n c i e s
a n d b a n d w i d t h s m u s t b e e q u a t e d w i t h the v a l u e s o b t a i n e d f r o m
3 . 2 . 2 6 a n d 3.2.27.
Thus f r o m 3.2.6 w i t h oip^ e q u a l to u o ^
we obtain
tan(acoo )[Z2 coo 2 Cl C2 -1]
r
r r r r
=
uo Z [Cl
r r r
+ C2 ]
r
(3.2.28)
The b a n d w i d t h of t h e rth r e s o n a t o r can e a s i l y be s h o w n t o b e
to a g o o d a p p r o x i m a t i o n
-1
Ao)r
=
[ 1 / Yr( ju) ]]
UJ = 0)0
(3.2.29)
T h u s f r o m 3 . 2 . 3 a n d 3.2.29 w e h a v e
Au)
r
=
K
r
F r o m 3.2.4 a n d 3 . 2 . 3 0 w e o b t a i n
(3.2.30)
47
AtD = wo Cl [Z mo C2 + tan(acoO ) ]
r
r r r r r
r
a(l+tan2 (au)0 ))[Z2 co0 2 Cl C2 -1] + 2tan (awo ) Z 2 u>o Cl C2 -Z (Cl -K2 )
r
r r r r
r r r r r r r r '
(3.2.31)
To o b t a i n the r e s o n a t o r e l e m e n t v a l u e s , f i r s t l y a n y o n e of
the variables
Z , Cl a n d C2 can b e p r e s c r i b e d to h a v e a
r
r
r
k n o w n v a l u e e q u a l fo r all r, e q u a t i o n s 3.2.28 a n d 3 . 2 . 3 1
c a n t h e n b e s o l v e d s i m u l t a n e o u s l y for t h e r e m a i n i n g v a r i a b l e s .
This p r o c e d u r e can e a s i l y be a c h i e v e d n u m e r i c a l l y ,
for example, the New to n- Rap hs on procedure
using
[3.2].
The p h a s e s h i fts b e t w e e n the r e s o n a t o r s a t t h e s t o p ­
b a n d c e n t r e f r e q u e n c y c a n n o w be e v a l u a t e d .
Firstly,
remember2
ing t h a t t he rth r e s o n a t o r has b e e n s c a l e d b y a f a c t o r n ^ ,
we obtain
K
/ Aa)
n 2 /A
r
r
r
Then compute
(3.2.32)
from
<ijz / t a n ( a m z )
r
r
- 1/(Z C2 )
r r
(3.2.33)
which can easily be solved numerically.
N ex t we obtain B
r
from
(3.2.34)
with
B
r
B'
r
+ B"
r
(3.2.35)
48
And
BT=,rr ++ 11 '
_=
Br '
(3.2.36)
R e a r r a n g i n g 3.2. 1 5 we o b t a i n t h e p h a s e s h i f t b e t w e e n the
resonators.
6r , r + l
=
c o t - 1 (B r " )
(3.2.37)
F r o m 3.2.16 w i t h K^. r + -^ = 1 (for the i n v e r s e C h e b y s h e v p r o t o ­
type)
the r e c u r r e n c e f o r m u l a e for t h e s c a l i n g f a c t o r s n ^ are
obtained.
n r+l
=
Sin(0r , r + l )/nr
(3.2.38)
With initial conditions
n± =
1
(3.2.39)
B1I =
0
(3.2.40)
In s u m m a r y , w e a p p l y 3 . 2.18 to 3 . 2 . 3 1 to o b t a i n t h e r e s o n a t o r
e l e m e n t v a l u e s a n d t h e n c a l c u l a t e the p h a s e s h i f t s f r o m 3 . 2.32
to 3.2.40.
It is o f i m p o r t a n c e t o n o t e t h a t the s c a l i n g p r o c e d u r e
u s e d in this d e s i g n p r o c e s s w i l l m o d i f y the b a n d w i d t h s o f t h e
resonators.
However, t h e c h a n g e in b a n d w i d t h is n e g l i g i b l e .
T h i s is b e c a u s e th e c o r r e c t p h a s e s h i f t b e t w e e n the r e s o n a t o r s
is a l w a y s g r e a t e r t h a n 75° in p r a c t i c e , a n d f r o m 3 . 2 . 3 8 w e
have
n
r
.n
..
r+1
=
sin (0
,.)
r,r+l
49
Thus for 0
n
,, > 75° then
r ,r+l
r
.n
,,
r+1
>
0.966
The e f f e c t of the s c a l i n g fact o r s on the r e s o n a t o r b a n d w i d t h s
is t h u s o f l i t t l e i m p o r t a n c e .
3.3
Design of Fixed Frequency Symmetrical Bandstop Filters
3.3.1
S p e c i f i c a t i o n s o f the f i l ters
T w o S . S . S b a n d s t o p filt e r s w e r e d e s i g n e d to t h e f o l l o w ­
ing specifications.
Design 1
Centre frequency
4 GHz
M i n i m u m s t o p b a n d r i p p l e level
40 dB
S t o p b a n d b andwidth
40 MHz
20 dB
66 MHz
r e t u r n loss b a n d w i d t h
Design 2
Centre frequency
4 GHz
M i n i m u m s t o p b a n d r i p p l e level
40 dB
Stopband bandwidth
400 MHz
20 dB r e t u r n loss b a n d w i d t h
660 M H z
The p h y s i c a l d e s i g n a n d c o n s t r u c t i o n o f e a c h o f t h e s e
f i l t e r s w a s v e r y s i m i l a r , thus o n l y D e s i g n 1 w h i c h is t h e m o r e
i m p o r t a n t , n a r r o w b a n d s p e c i f i c a t i o n is p r e s e n t e d in detail.
50
3.3.2
P h y s i c a l D e s i g n of the B a n d s t o p F i l t e r s
Design 1
The t h e o r e t i c a l d e s i g n p r o c e s s d e s c r i b e d in s e c t i o n
3.2 e n a b l e s one set of the r e s o n a t o r e l e m e n t v a l u e s , Cl ,
'
r'
C 2 ^ a n d Z^ to b e p r e s c r i b e d .
In the case o f f i x e d f r e q u e n c y
f i l t e r s it is n o t n e c e s s a r y for the t u n i n g c a p a c i t o r s to be
equal,
thus e q u a t i o n s 3.2. 2 8 a n d 3 . 2.31 s h o u l d b e s o l v e d
s i m u l t a n e o u s l y f o r C l ^ a n d C2^.
In the p a r t i c u l a r c a s e o f
this d e s i g n w h i c h is f o r a 1% s t o p b a n d b a n d w i d t h a t 4 GHz
t h e r e w a s n o real n e e d to s olve f o r Cl
s i n c e t h e 3 dB
band-
w i d t h s of th e r e s o n a t o r s c o u l d b e u s e d d i r e c t l y to o b t a i n
the c o u p l i n g g a p s f r o m the r e s o n a t o r s to the m a i n l i n e f r o m
Fig. 2.1.9.
T h e p h a s e l e n g t h s of the r e s o n a t o r s u s e d in this
f i l t e r w e r e d e s i g n e d t o be 160° at 4 GHz
and the resonator
c h a r a c t e r i s t i c i m p e d a n c e s w e r e all c h o s e n to b e 5 0 Q .
The
m e t h o d d e s c r i b e d in s e c t i o n 3.2 w a s then a p p l i e d to c a l c u l a t e
t h e 3 dB
b a n d w i d t h s of the r e s o n a t o r s a n d t h e p h a s e s h i f t s
b e t w e e n them.
This p r o c e d u r e a l s o e v a l u a t e d t h e v a l u e s o f
t h e t u n i n g c a p a c i t o r s C 2 r , these w e r e all f o u n d t o b e less
t h a n 0 . 2 5 p F w h i c h are e a s i l y r e a l i s a b l e u s i n g c a p a c i t i v e t u n i n g
screws.
The v a l u e s o f the r e s o n a t o r b a n d w i d t h s a n d t h e p h a s e
s h i f t s b e t w e e n t h e m are s h o w n in Fig.
3.3.1.
Th e c o n v e r s i o n f r o m e l e c t r i c a l to p h y s i c a l p a r a m e t e r s
is n o w e a s i l y p e r f o r m e d .
First, a g r o u n d p l a n e s p a c i n g of
0 . 1 5 " w a s c h o s e n , p r e v i o u s r e s u l t s i n d i c a t e d t h a t this w o u l d
Resonator
number
Bandwidth
of
resonator
MHz
r,r+l
Fig.
3.3.1
1
2
9
25
84.6
87.4
3
4
5
6
7
35
40.5
35
25
9
83.7
83.7
87.4
84.6
T he c o m p u t e d v a l u e s of the b a n d w i d t h s of the
p h a s e s h i f t s b e t w e e n the r e s o n a t o r s of the
narrow band bandstop filter
51
g i v e a r e a s o n a b l e v a l u e of r e s o n a t o r Q factor.
S e c o n d l y f r o m Fig.
3.3.1 a n d Fig.
2.1.9 the c o u p l i n g
gaps b e t w e e n the r e s o n t o r s a n d the m a i n line are o b t a i n e d .
The w i d t h s o f the 50ft
r e s o n a t o r s in a 0 . 1 5 " d e e p
c a v i t y w e r e c a l c u l a t e d to b e 0 . 2 1 3 " u s i n g t h e m e t h o d d e s c r i b e d
in s e c t i o n 2.1.2.
The p h y s i c a l l e n g t h s o f the m a i n line s e p a r a t i n g the
r e s o n a t o r s are c a l c u l a t e d as follows.
A t the s t o p b a n d c e n t r e f r e q u e n c y wo, w e h a v e
_
'
r, r+l
laoLm
, , , ,x
(3.3.1)
where
Lm
=
l e n g t h o f m a i n line b e t w e e n r e s o n a t o r s
c
=
v e l o c i t y o f e l e c t r o m a g n e t i c p r o p o g a t i o n in air.
The r e f e r e n c e p l a n e l o c a t i o n s b e t w e e n t h e r e s o n a t o r s
m u s t a l s o b e t a k e n i n t o a c c o u n t in this c a l c u l a t i o n .
From
3 . 3 . 1 a n d Fig. 2 . 2 . 1 7 w e h a v e
Lm'
=
Cib
, .
r rr±l
0)0
+
u
r
io
+
oj
. ,
r+l
io
(3
where
w
r
=
w i d t h o f rth r e s o n a t o r
E q u a t i o n 3.3.2 e n a b l e s the c a l c u l a t i o n o f t h e e x a c t
r e q u i r e d len g t h of m a i n line b e t w e e n the c e n t r e lines o f t h e
resonators.
The p h a s e l e n g t h s of the r e s o n a t o r s w e r e a l l r e q u i r e d
t o b e 160° a t 4 G H z .
This c o r r e s p o n d s t o a p h y s i c a l l e n g t h
52
of 1.319".
The r e f e r e n c e p l a n e l o c a t i o n s o f the r e s o n a t o r s
due to f r i n g i n g a t o ne e n d a n d t h e d i s c o n t i n u i t y at t h e
coupling region must be taken into account.
From 2.1.14
the r e q u i r e d p h y s i c a l length of e a c h r e s o n a t o r is
LRESr
=
1.319" - u r - 0 . 2 3 b -
(3.3.3)
where
ur
=
the w i d t h of t he rth c o u p l i n g gap
wm
=
th e w i d t h o f the m a i n line
b
=
th e g r o u n d p l a n e s p a c i n g
The m a i n line w a s to b e l o c a t e d at a d i s t a n c e 0 .05"
f r o m the i n n e r side w a l l of the S . S . S h o u s i n g ,
the w i d t h o f
this line a f t e r a l l o w i n g for the i n c r e a s e in f r i n g i n g
c a p a c i t a n c e to the side w all, w a s c a l c u l a t e d u s i n g t h e m e t h o d
d e s c r i b e d in s e c t i o n 2 .1.2 a n d was
wm
=
0 . 209"
The l e n g t h s of the r e s o n a t o r s w e r e thus
LRES
r
=
1.23 2" -u
(3.3.4)
r
The p h y s i c a l l a y o u t of the S . S . S c i r c u i t b o a r d f o r t h e b a n d ­
s t o p f i l t e r is s hown in Fig.
3.3.2.
D e s i g n of t he h o u s i n g for the b a n d s t o p f i l t e r
The d e s i g n of the S . S . S h o u s i n g was v e r y s i m i l a r to
t h a t d e s c r i b e d in c h a p t e r 2.
The m a j o r d i f f e r e n c e w a s t h a t
Fig.
3.3.2
The
dimensions
of
the
circuit
board
of
the
narrow
band
r~i
rsi
o
i
co
rsi
filter
l.in
bandstop
0>
O
(N
o
53
due to the l e n g t h of the h o u s i n g , s p u r i o u s m o d e p r o p o g a t i o n
could be a severe problem.
The a c t u a l l e n g t h of the i n t e r i o r
of t h e h o u s i n g w a s 5.11".
This w o u l d e n a b l e a T E 1 0 1 mo<^e
t o r e s o n a t e at 3.65 GHz.
To o v e r c o m e this p r o b l e m ,
inductive
p o s t s w e r e i n s e r t e d b e t w e e n t h e r e s o n a t o r s , so t h a t t h e t o p
o f th e h o u s i n g w a s e l e c t r i c a l l y c o n n e c t e d to t h e b o t t o m
[Fig.
3.3.3].
These posts were ap pr oximately one quar te r
w a v e l e n g t h a p a r t a t 4 GHz
and wo u l d raise the b o x resonant
f r e q u e n c y to 8 . 7 1 GHz.
T u n i n g s c r e w s w e r e l o c a t e d in the top o f t h e h o u s i n g
so t h a t t h e y w e r e o v e r the e n d s o f t h e r e s o n a t o r s .
An
i m p o r t a n t p r a c t i c a l p o i n t to n o t e is t h a t t h ese s c r e w s w e r e
p o s i t i o n e d a d j a c e n t to n y l o n plugs, this gave t h e m m o r e
r i g i d i t y a n d m a d e fine t u n i n g r e l a t i v e l y simple.
The S . S . S h o u s i n g w a s c o p p e r p l a t e d to r e d u c e its
loss a n d the a s s e m b l e d f i l t e r is s h o w n in p h o t o g r a p h n o 1.
Design 2
Th e o n l y s i g n i f i c a n t d i f f e r e n c e b e t w e e n t h i s d e s i g n
a n d d e s i g n 1 w a s t h a t the l a r g e r b a n d w i d t h o f t h e f i l t e r
r e s u l t e d in v a l u e s of c o u p l i n g c a p a c i t o r s Cl^. to t h e m a i n line
w h i c h w e r e to o l a rge to b e r e a l i s e d w i t h p r a c t i c a l v a l u e s o f
c o u p l i n g gaps.
To o v e r c o m e this p r o b l e m it w a s n e c e s s a r y to
p r i n t th e r e s o n a t o r s on o p p o s i t e s ides to a n d o v e r l a p p i n g t h e
m a i n line [Fig.
3.3.4].
T h e c o u p l i n g c a p a c i t o r s w e r e then
realised by parallel plates.
No e x a c t p r o c e d u r e is k n o w n f o r
c a l c u l a t i n g the o v e r l a p r e q u i r e d to p r o d u c e a d e s i r e d
capacitance.
In p r a c t i c e t h e c a p a c i t o r w a s a s s u m e d t o be a
P h o t o g r a p h no.
1
4 GHz B a n d s t o p F i l t e r
conductor
Fig.
3 .3.3
S h o w i n g the m e t h o d of s u p p r e s s i n g h i g h e r
o r d e r e d m o d e s in the b a n d s t o p f i l t e r
Fig.
3 . 3.4
O v e r l a p p i n g c a p a c i t o r r e a l i s a t i o n of l arge
capacitance
54
a s i m p l e p a r a l l e l p l a t e c a p a c i t o r t h r o u g h the d i e l e c t r i c
c i r c u i t b o a r d [er = 2.22] a n d an i n c r e a s e in c a p a c i t a n c e of
50% due to f r i n g i n g w a s assumed.
The d e t a i l s o f t h e d e s i g n
of this f i l t e r are n o t p r e s e n t e d b e c a u s e o f the s i m i l a r i t y
to d e s i g n 1.
3 . 3.3
M e a s u r e d P e r f o r m a n c e s of the B a n d s t o p F i l t e r s
The m e a s u r e d f r e q u e n c y r e s p o n s e s o f t h e t w o b a n d s t o p
f i l t e r s are s h o w n in Figs 3.3.5 a n d 3.3.6.
The p e r f o r m a n c e s
are s u m m a r i s e d below.
Design 1
Stopband centre frequency
4 GHz
1 dB
stopband bandwidth
90 MHz
3 dB
stopband bandwidth
62.5 MHz
40dB s t o p b a n d b a n d w i d t h
40 MHz
M i d - s t o p b a n d r e t u r n loss
1 dB
M i n i m u m p a s s b a n d r e t u r n loss
14 dB
to 7 GHz
Design 2
Stopband centre frequency
4 GHz
1 dB
stopband bandwidth
670 M H z
3 dB
stopband bandwidth
600 M H z
40 dB s t o p b a n d b a n d w i d t h
400 M H z
M i d - s t o p b a n d r e t u r n loss
0.4 dB
M i n i m u m p a s s b a n d r e t u r n loss
14 dB
to 7 GHz
cn
k
Fig.
3.3.5
Measured
performance
of
narrowband
bandstop
filter
Fig.
3.3.6
Measured
performance
of
10%
bandwidth
bandstop
filter
55
B o t h f i l t e r s e x h i b i t e d the r e q u i r e d s y m m e t r i c a l
f r e q u e n c y r e s p o n s e a n d a h i g h p a s s b a n d r e t u r n loss.
The
g o o d p a s s b a n d r e t u r n loss w a s a c o n s e q u e n c e o f u s i n g a
s t r a i g h t t h r o u g h t r a n s m i s s i o n line f o r c o u p l i n g the r e s o n a t o r s .
The 10% b a n d w i d t h f i l t e r e x h i b i t e d a l m o s t t h e e x a c t d e s i g n e d
specification.
H o w e v e r the 1% b a n d w i d t h f i l t e r s u f f e r e d f r o m
a r e d u c t i o n of s e l e c t i v i t y due t o the f i n i t e Q f a c t o r of
the S.S.S medium.
Th e d e s i g n e d p a s s b a n d t o 40 dB
stopband
r a t i o of this f i l t e r w a s 1.66:1, in p r a c t i c e , the 1 dB
40 dB
r a t i o w a s 2 . 2 5 : 1 a n d t h e 3 dB
to 40 dB
to
r a t i o w a s 1.5:1.
The s e l e c t i v i t y of this f i l t e r c o u l d b e i n c r e a s e d b y i n c r e a s ­
i ng the g r o u n d p l a n e s p acing,
Q factors.
However,
thus i n c r e a s i n g t h e r e s o n a t o r
t h e re is o b v i o u s l y a l i m i t b e y o n d w h i c h
it is n o t p r a c t i c a b l e to i n c r e a s e the g r o u n d p l a n e s p a c i n g .
F o r e x a m p l e , c o n s i d e r th e case of a f i l t e r w i t h 50ft
resonators.
The w i d t h o f th e r e s o n a t o r s is g i v e n b y
F o r 50ft
r e s o n a t o r s c/e = 7.54 a n d for large g r o u n d p l a n e
cf'
spacings — —
w
= 0.45,
thus
(3.3.6)
1.43 b
Th e p h a s e s h i f t b e t w e e n the r e s o n a t o r s m u s t b e a p p r o x i m a t e l y
90° a n d th e p h y s i c a l d i s t a n c e b e t w e e n the e d g e s of the
r e s o n a t o r s is
s
1.43 b
(3.3.7)
56
w h e r e c is th e v e l o c i t y of e l e c t r o m a g n e t i c p r o p o g a t i o n in
a i r a n d s a n d b are in inches.
T h e v a l u e of s m u s t h a v e a m i n i m u m v a l u e s i n c e if the
r e s o n a t o r s a r e too c l o se to e a c h o t h e r t h e r e w i l l b e s i g n i f i ­
c a n t c o u p l i n g b e t w e e n them.
Experimental results have shown
t h a t th e v a l u e of s s h o u l d s a t i s f y the i n e q u a l i t y
s
>
0 . 3b
(3.3.8)
T e s e t s on f i l t e r s n o t s a t i s f y i n g this i n e q u a l i t y s h o w that
the s t o p b a n d r e s p o n s e m a y d e t e r i o r a t e to the p o i n t of b e c o m i n g
an al l pass r e s p o n s e .
F r o m 3 . 3.7 an d 3 . 3 . 8 w e o b t a i n
buo
<
1.07 x 1 0 10
(3.3.9)
T h u s fo r a s t o p b a n d c e n t r e f r e q u e n c y of 4 G H z w e o b t a i n
b
<
0.4 2 5"
It is i m p o r t a n t to n o t e that the i m p e d a n c e s o f the r e s o n a t o r s
c o u l d be i n c r e a s e d b e y o n d 50ft,
this w o u l d r e d u c e t h e i r w i d t h
a n d thus the g r o u n d p l a n e s p a c i n g c o u l d b e i n c r e a s e d .
However
t h e Q f a c t o r of S . S . S t r a n s m i s s i o n lines d e c r e a s e s l i n e a r l y
w i t h i m p e d a n c e an d thus n o a d v a n t a g e w o u l d b e g a i n e d f r o m
d o i n g this.
3.3.4
C o m p u t e r A n a l y s i s of S .S.S F i x e d F r e q u e n c y F i l t e r s
A c o m p u t e r p r o g r a m w a s w r i t t e n to a n a l y s e the e f f e c t
of r e s i s t i v e loss on the p e r f o r m a n c e of f i x e d f r e q u e n c y S . S . S
57
b a n d s t o p filters.
In this p r o g r a m t h e m a i n line and
r e s o n a t o r t r a n s m i s s i o n lines w e r e r e p r e s e n t e d b y the g e n e r a l
t r a n s f e r m a t r i x for a lossy t r a n s m i s s i o n line, w h i c h is
cosh
(yL)
Zo sin h (y L )
(3.3.10)
cosh
Y o sin h (yL)
(yL)
w h e r e to a g o o d a p p r o x i m a t i o n f o r low loss lines [3.3]
y
=
a + jB
(3.3.11)
a
=
g/2Qc
(3.3.12)
3
=
cj/c
(3.3.13)
Qc is the q u a l i t y f a c t o r o f t h e S . S . S t r a n s m i s s i o n lines
c o n t r i b u t e d b y c o n d u c t o r losses [Fig.
3.3.7].
[3.4]
The
d i e l e c t r i c l o s s e s of the S .S.S lines w e r e a s s u m e d to b e i n ­
significant.
The c o m p u t e d r e s p o n s e o f the 1% b a n d w i d t h f i l t e r
d e s c r i b e d in s e c t i o n 3.3.3 is s h o w n in Fig.
excellent agreement
3.3.8, this s h o w s
w i t h the m e a s u r e d p e r f o r m a n c e o f t h e
f ilter.
B e c a u s e o f th e c l o s e a g r e e m e n t b e t w e e n t h e c o m p u t e d
a n d m e a s u r e d p e r f o r m a n c e of the b a n d s t o p filter, it w a s
p o s s i b l e t o us e the p r o g r a m to p r e d i c t t h e p e r f o r m a n c e o f a n y
S . S . S b a n d s t o p filter.
a n d 3 dB
to 40 dB
Fig.
3.3.9 s h ows the 1 dB
to 40 dB
b a n d w i d t h r a t i o s of b a n d s t o p f i l t e r s f r o m
d e g r e e 5 t o d e g r e e 9, w i t h b a n d w i d t h s f r o m 0 . 5 % to 5%.
This
l
1
i—i
i
ij
I
I
O
Zo / er - ohms
Fig.
3 . 3.7
The n o r m a l i s e d Q f a c t o r of an S . S . S t r a n s m i s s i o n
line as a f u n c t i o n of c h a r a c t e r i s t i c i m p e d a n c e .
T h i s r e f e r s to a zero t h i c k n e s s b a r n o r m a l i s e d
t o g r o u n d p l a n e s p a c i n g and f r e q u e n c y
Fig.
3.3.8
Computed
response
of
narrowband
S.S.S
bands too
filter
V)
9 8
tn
s
e
Au
%
Fig.
Degree 5
Degree 7
Degree 9
1 : 40
3 : 40
1 : 40
3 :4
1 : 40
3 :40
0.5
3.45
2.12
2.54
1.72
2.50
1.62
1
2.24
2.03
1.95
1.59
1.83
1.41
2
2.12
1.98
1.63
1.5
1.45
1.31
5
1.91
1.83
1.54
1.48
1.33
1.28
10
1.74
1.68
1.46
1.41
1.26
1.21
3.3.9
S h o w i n g the c o m p u t e d 1 dB
to 40 dB
to 40 dB
and 3 dB
b a n d w i d t h ratios of S . S . S b a n d s t o p
f i l t e r s w i t h 0.3" g r o u n d p l a n e s p a c i n g and
copper conductors
58
i n f o r m a t i o n w a s a l l d e r i v e d f r o m the a n a l y s i s of 4 GHz
filters
w i t h 0.3" g r o u n d p l a n e spacings, h o w e v e r , s i n c e t h e f i l t e r Q
f a c t o r is l i n e a r l y p r o p o r t i o n a l to g r o u n d p l a n e s p a c i n g a n d
p r o p o r t i o n a l to the s q u a r e r o o t o f f r e q u e n c y ,
it is p o s s i b l e
t o p r e d i c t th e p e r f o r m a n c e of v i r t u a l l y any f i l t e r b y s i m p l e
extrapolation.
3.4
V a r a c t o r T u n e d S.S.S B a n d s t o p F i l t e r s
3.4.1
Computer Analysis of Varactor Tuned Filters
In c h a p t e r t w o it w a s seen t h a t t h e p e r f o r m a n c e o f
v a r a c t o r t u n e d f i l t e r s w a s c r i t i c a l l y d e p e n d e n t on t h e Q
f a c t o r of th e v a r a c t o r s
(which is low at m i c r o w a v e f r e q u e n c i e s )
a n d t h e s p e c i f i c a t i o n o f the filter.
It w a s thus n e c e s s a r y
to b e a b l e t o p r e d i c t a c c u r a t e l y the p e r f o r m a n c e o f any
v a r a c t o r t u n e d f ilter.
This w a s a c h i e v e d b y c o m p u t e r a n a l y s i s
o f th e f i l t e r s w i t h th e v a r a c t o r s m o d e l l e d b y t h e i r e q u i v a l e n t
ci r c u i t .
[Fig.
3.4.1]
The p a r t i c u l a r v a r a c t o r to b e u s e d in t h e r e m a i n d e r
of t h e w o r k d e s c r i b e d in this t h esis w a s t',e M A 4 6 6 2 0 G
manufactured by Microwave Associates.
device
This v a r a c t o r h a d t h e
h i g h e s t Q f a c t o r of a n y p r e s e n t l y a v a i l a b l e d e v ice,
its
e q u i v a l e n t c i r c u i t p a r a m e t e r s w e r e c l a i m e d t o b e as f o l l o w s
C j (0)
=
0.8pF + 0.04pF
C j (60V)
=
0 . 12pF + 0.02pF
Rj (v)
=
£ 1. 5ft
Cpl
=
0 . 19pF + 0.02pF
Cp2
=
0.04pF + O.OlpF
Lp
=
400pH + 50p H
Lp
Rj (v)
Cj (v)
Fig.
3.4.1
MA 46620G varactor equivalent circuit
59
T h e s e p a r a m e t e r s w e r e c h e c k e d u s i n g the m e a s u r e m e n t
t e c h n i q u e s d i s c u s s e d in s e c t i o n 2.2.2.
T h e m e a s u r e d v alue
of b o n d w i r e i n d u c t a n c e of 1 4 0 0 p H c o n t r a d i c t e d the m a n u f a c t ­
u r e r s c l a i m of 400pH.
It w o u l d thus s e e m w i s e to b e w a r y
of manufacturers claims and always measure varactor propert­
ies b e f o r e p r o c e e d i n g w i t h a design.
A c o m p u t e r a n a l y s i s p r o g r a m w a s w r i t t e n in o r d e r
to p r e d i c t the p e r f o r m a n c e o f v a r a c t o r t u n e d f i l t e r s as a
f u n c t i o n of d e g r e e a n d b a n d w i d t h .
The c i r c u i t m o d e l o f the
M A 4 6 6 2 0 G v a r a c t o r w a s u s e d in the a n a l y s i s .
The varactor
t u n e d f i l t e r s w e r e a n a l y s e d for b a n d w i d t h s f r o m 0.5% to 5%
a n d d e g r e e s of 2, 3 a n d 4 for v a l u e s o f Cj o f 0.8p F , 0 . 4 p F
a n d O .llpF.
Th e c o m p u t e d ldB to 20dB and 3dB t o 2 0 d b b a n d ­
w i d t h r a t i o s of the f i l t e r s are p r e s e n t e d in Fig.
3.4.2. A l l
t he d a t a p r e s e n t e d w a s for f i l t e r s w i t h d i r e c t v a r a c t o r
c o u p l i n g to the r e s o n a t o r s , c o r r e s p o n d i n g to a t u n i n g b a n d ­
w i d t h o f 25%.
Fig.
Th e b a n d w i d t h s o f the f i l t e r s p r e s e n t e d in
3.4.2 ar e the 2 0 d B b a n d w i d t h s at t h e m i d d l e o f t h e t u n i n g
b a n d , t h e s e v a r i e d b y a f a c t o r of 1.6:1 a c r o s s the t u n i n g
band.
The c e n t r e f r e q u e n c y o f all the f i l t e r s s h o w n in Fig.
3.4.2
w a s 4GHz.
In o r d e r to p r e d i c t the p e r f o r m a n c e o f f i l t e r s
w i t h o t h e r c e n t r e f r e q u e n c i e s the c h a n g e in f i l t e r Q f a c t o r
w i t h f r e q u e n c y m u s t b e t a k e n i n t o acco u n t .
The Q f a c t o r of
th e v a r a c t o r s is i n v e r s e l y p r o p o r t i o n a l t o f r e q u e n c y a n d the
Q f a c t o r o f the S . S . S m e d i u m is too large to be s i g n i f i c a n t .
Thus, s i n c e the Q f a c t o r o f the f i l t e r is p r o p o r t i o n a l to its
bandwidth,
the a n a l y s i s of a n y f i l t e r can b e a c h i e v e d s i m p l y
20 dB
Select­
bandwidth ivity
%
1:20 dB
0.5
Fig.
Select­
ivity
3:20 dB
Cj = 0.2 pF
Cj = 0 . 4 pF
Cj = 0.8 pF
Select­
ivity
1:20 dB
Select­
ivity
3:20 dB
Select­
ivity
1:20 dB
13
9
4.33
Not ap| licable
since £topband
rejection only 12 <£
Select­
ivity
3:20 dB
3.16
1
14.01
9.5
4.66
3.44
4.13
2.95
2
4.50
3.31
3,90
3.12
3.46
2.84
5
3.19
2.70
3.05
2.58
3.13
2.66
3.4.2a
C o m p u t e d p e r f o r m a n c e of s e c o n d d e g r e e v a r a c t o r
t u n e d b a n d s t o p filters.
t o 20 dB
and 3 dB
T h i s shows the 1 dB
to 20 dB
s e l e c t i v i t i e s of
the f i l t e r s as a f u n c t i o n of v a r a c t o r c a p a c i t ­
ance and
stopband bandwidth
Cj = 0.8 pF
20 dB Select­
bandwidth ivity
%
1:20 dB
0.5
Fig.
Select­
ivity
3:20 dB
Not ap| ilicable
since maximum
rejecticn cnly B.7cB
Cj = 0.4 pF
Cj = 0.2 pF
Select­
ivity
1:20 dB
Select­
ivity
3:20 dB
Select­
ivity
1:20 dB
6.92
4,05
3.91
2.48
Select­
ivity
3:20 dB
1
8.55
4.73
4.20
2.66
3.04
2.15
2
5.03
2.71
3.44
2.20
2.58
2.08
5
3.19
2.00
2.14
1.81
1.97
1.73
3.4.2b
C o m p u t e d p e r f o r m a n c e of t h i r d d e g r e e v a r a c t o r
t u n e d b a n d s t o p filters.
to 20 d B
and 3 dB
T h i s s h o w s the 1 d B
to 20 d B
s e l e c t i v i t i e s of
the f i l t e r as a f u n c t i o n of v a r a c t o r c a p a c i t a n c e
an d s t o p b a n d b a n d w i d t h .
Cj = 0 . 8 pF
20 dB
Select­
bandwidth ivity
%
1:20 dB
0.5
Fig.
Select­
ivity
3:20 dB
Not app Licable
since m axirrum
rejection only 15 dE
Cj = 0 . 4 pF
Cj = 0.2 pF
Select­
ivity
1:20 dB
Select­
ivity
3:20 dB
Select­
ivity
1:20 dB
Select­
ivity
3:20 dB
5.10
3.40
2.64
2.05
1
6.43
3.95
2.88
2.16
2.29
1.83
2
3.41
2.52
2.21
1.89
2.07
1.64
5
2.25
1.85
1.76
1.61
1.70
1.53
3.4.2c)
C o m p u t e d p e r f o r m a n c e of f o u r t h d e g r e e v a r a c t o r
t u n e d b a n d s t o p filters.
T h i s s hows the 1 dB
to 20 dB and 3 dB to 20 dB s e l e c t i v i t i e s of
the f i l t e r as a f u n c t i o n of v a r a c t o r c a p a c i t a n c e
an d s t o p b a n d b a n d w i d t h
60
b y d i v i d i n g the b a n d w i d t h b y a f a c t o r e q u a l t o the i n c r e a s e
o r d e c r e a s e in f r e q u e n c y f r o m 4 GHz.
3.4.2
D e s i g n of a V a r a c t o r T u n e d B a n d s t o p F i l t e r
The s p e c i f i c a t i o n of t h e v a r a c t o r t u n e d f i l t e r to
be constructed was
Centre frequency
4 GHz
20 dB
50 M H z
stopband bandwidth
+ 500 MHz
at 4 GHz
Degree
3
P a s s b a n d r e t u r n loss
20 dB
The c o m p u t e d p e r f o r m a n c e of this f i l t e r is s h o w n in
F i g s . 3 .4.3 - 3.4.5.
The d e s i g n o f this f i l t e r w a s b a s e d u p o n t h e p r o c e d ­
u r e s o u t l i n e d in s e c t i o n s 3.2 to 3.3.3.
The only significant
d i f f e r e n c e s in the d e s i g n of this f i l t e r as c o m p a r e d to t h e
f i x e d f r e q u e n c y f i l t e r s w e r e as follows.
C h o i c e of P r o t o t y p e
Th e d e s i g n w a s b a s e d on the C h e b y s h e v p r o t o t y p e filt e r .
This w a s p u r e l y f or th e p u r p o s e s o f e a s e o f c o m p u t e r a n a l y s i s .
Since the
i n v e r s e C h e b y s h e v p r o t o t y p e f i l t e r has t r a n s m i s s i o n
z e ros at i n t e r v a l s a c r o s s the s t o p b a n d of the filter, as t h e
c o m p u t e r a n a l y s i s p r o g r a m t u n e d the filter, t h e t r a n s m i s s i o n
zeros b e c a m e m i s a l i g n e d a n d the r e s p o n s e b e c a m e m e a n i n g l e s s .
Th e t r a n s m i s s i o n zeros o f the C h e b y s h e v f i l t e r are all at t h e
same frequency a n d the p r o b l e m does n o t o c c u r w h e n u s i n g this
prototype.
The m o d i f i c a t i o n to the d e s i g n p r o c e s s i n v o l v e d in
Fig.
3.4.3
Computed
performance
o:
fC
*H
O
M
0)
-P
N
of
practical
varactor
tuned
bandstop
f i lter
Fig.
-100MHz
3.4.4
-4V
-75 MHz
at
Computed
-50 MHz
bias
of
-25 MHz
performance
4.03 GHz
practical
+25 MHz
tuned
+50 MHz
varactor
+75 MHz
bandstop
+100MHz
f i lter
Fig.
-75 MHz
3.4.5
-50 MHz
filter
Computed
at
-25 MHz
-30V
bias
performance
tuned
+50 MHz
varactor
+25 MHz
practical
4.54 GHz
of
+75 MHz
bandstop
c/i
5 I s e
m
61
u s i n g t h i s p r o t o t y p e is trivial,
it m e r e l y i n v o l v e s u s i n g
t he e l e m e n t v a l u e s g i v e n in ref.
3.5.
It is i m p o r t a n t to n o t e t h a t w h e n d e s i g n i n g t h e
e l e m e n t v a l u e s of the t u n a b l e b a n d s t o p filter,
c a p a c i t o r s m u s t be c h o s e n a n d then e q u a t i o n s
the t u n i n g
3.2.28 a n d
3 . 2 . 2 9 s h o u l d b e s o l v e d for the c h a r a c t e r i s t i c i m p e d a n c e s of
t h e r e s o n a t o r s a n d the c o u p l i n g c a p a c i t a n c e s to t h e m a i n
line.
Compensation for Varactor Inductance
The v a r a c t o r c a p a c i t a n c e ,
including package capacit­
a n c e w a s lpF at z e r o b i a s a n d 0 . 3pF at 60 v o l t s bias.
It
w o u l d thus s e e m o b v i o u s t h a t the f i l t e r s h o u l d b e d e s i g n e d
f o r a m i d b a n d t u n i n g c a p a c i t a n c e e q u a l to t h e g e o m e t r i c m e a n
of these two values.
T h a t is 0.55pF.
H o w e v e r the b o n d w i r e
i n d u c t a n c e of t h e v a r a c t o r has the e f f e c t of l o w e r i n g the
c e n t r e f r e q u e n c y o f the f i l t e r .
To c o m p e n s a t e for
this the f i l t e r w a s d e s i g n e d for a m i d b a n d t u n i n g c a p a c i t o r
v a l u e o f 0. 6 5 p F .
The e l e m e n t v a l u e s of the t u n a b l e b a n d s t o p f i l t e r
w e r e as f o l l o w s
Resonator
Impedance of
resonator
Coupling capacitance
to m a i n line
1
76.5 ft
1.72 x 1 0 -13 F
2
76 ft
1.66 x I O - 1 3 F
3
76.5ft
1. 72 x I O -33 F
62
T h e p h a s e len g ths b e t w e e n the r e s o n a t o r s w e r e
c o m p u t e d to be 84.6° at 4 GHz and the p h a s e lengths of the
r e s o n a t o r s w e r e 125° at 4 GHz.
T h e g r o u n d p l a n e s p a c i n g of the c a v i t y was c h o s e n to
b e 0.3" an d the p h y s i c a l d i m e n s i o n s of the r e s o n a t o r s and
c o u p l i n g gaps w e r e c a l c u l a t e d u s i n g the m e t h o d s d e s c r i b e d
in s e c t i o n 3.2.2.
D e s i g n of the M a i n T h r o u g h L i n e
A m e t h o d of r e d u c i n g the w i d t h of the m a i n line w a s
d e s c r i b e d in s e c t i o n 2.1.2.
This i n v o l v e d l o c a t i n g the m a i n
l i n e n e a r to the s i d e w a l l so that the f r i n g i n g c a p a c i t a n c e
to g r o u n d w a s i n c r e a s e d .
In p r a c t i c e ,
this r e q u i r e s v e r y
a c c u r a t e a l i g n m e n t of the c i r c u i t b o a r d w h i c h m a y n o t a l w a y s
be achieved.
A n y e r r o r s in c i r c u i t b o a r d a l i g n m e n t w o u l d
r e s u l t in a c h a n g e in i m p e d a n c e of the m a i n line and a
c o n s e q u e n t d e g r a d a t i o n in the p a s s b a n d r e t u r n loss of the
f ilter.
A n a l t e r n a t i v e to this t e c h n i q u e is to p r i n t a
g r o u n d e d s t r i p v e r y c l o s e to the m a i n line as s h o w n in Fig.
3.4.6.
T he a d v a n t a g e of this t e c h n i q u e is t h a t s i n c e the
m a i n line w i l l a l w a y s be the s a m e d i s t a n c e to g r o u n d ,
i n d e p e n d e n c e of e r r o r s in c i r c u i t b o a r d loca t i o n ,
t h e n good,
r e p r o d u c i b l e p a s s b a n d r e t u r n loss levels can be a c h i e v e d .
A l s o , b e c a u s e p r i n t e d c i r c u i t s can be p r i n t e d to w i t h i n an
a c c u r a c y of 0 . 0 0 0 5 " , t h e n c o n s i d e r a b l e r e d u c t i o n in w i d t h of
th e m a i n line can be achieved.
In this p a r t i c u l a r c a s e the
w i d t h of th e m a i n line w a s r e d u c e d f r o m 0 . 4 " to 0 . 2 7 3 " by
Grounded strip
Fig.
3.4.6
G r o u n d e d s t r i p r e a l i s a t i o n of t h i n lines
strip
Fig.
3.4.7
D,C bias input
R e a l i s a t i o n of bias c h o k e
63
p r i n t i n g t h e g r o u n d e d s t r i p at a d i s t a n c e of 0.01" f r o m
th e m a i n line.
D e s i g n of B i a s C h o k e s
In s e c t i o n 2.2.3 the d e s i g n of s t e p p e d i m p e d a n c e
t r a n s m i s s i o n line b i a s chokes was d i s c u s s e d .
Unfortunately,
a t wo s e c t i o n q u a r t e r w a v e s t e p p e d i m p e d a n c e f i l t e r o p e r a t i n g
at 4 GH z w i l l be a p p r o x i m a t e l y 1.4" long.
It is d e s i r a b l e
t o m i n i m i s e th e p h y s i c a l size of d e v i c e s for use in m i l i t a r y
systems.
T h u s an a l t e r n a t i v e t e c h n i q u e is to u s e a s i n g l e
s e c t i o n of h i g h i m p e d a n c e q u a r t e r w a v e t r a n s m i s s i o n t e r m i n a t e d
in a s h u n t c a p a c i t o r to ground.
[Fig. 3.4.7]
This technique
h a l v e s the l e n g t h of the bias f i l t e r and r e m o v e s t h e n e e d to
r e a l i s e v e r y low i m p e d a n c e t r a n s m i s s i o n lines.
The p a r t i c u l a r
b i a s f i l t e r u s e d in t h is d e s i g n h a d a q u a r t e r w a v e f r e q u e n c y
of 4 GHz a n d an i m p e d a n c e of 200 ft , the s h u n t c a p a c i t o r w a s
a l O p F c h i p c a p a c i t o r m a n u f a c t u r e d by A.T.C.
capacitors and
this w a s g u a r a n t e e d n o n - r e s o n a n t for f r e q u e n c i e s b e l o w 10 GHz.
T h e t r a n s f e r m a t r i x of the b i a s c hoke is
c o s (e )
jY si n ( 0)
j Zsin (0)
1
c o s (0)
jwC
—
—
—
0
1
—
—
cos (0)
j[Ysin(0)
—
-u)CZsin(0)
+
ioCcos(0)
j Z s i n (0)
c o s (0)
__
64
The i n s e r t i o n loss of this f i l t e r o p e r a t i n g in a
lft s y s t e m is e x p r e s s e d by
LA = lDLog[l+0.25C (o)(Zsin(e))2 + ((Z-Y) sin (0) - io(cos (0)) 2 ]]
(3.4.3)
w h e r e C a n d Z are n o r m a l i s e d to a 1ft system.
E v a l u a t i n g 3 .4.3 at 4 GHz
L A (4 GH z )
=
we o b t a i n
2 8.6 dB
This f i l t e r t h u s p r o d u c e s a s u p e r i o r i n s e r t i o n loss to t h e
s t e p p e d i m p e d a n c e f i l t e r a n d is o n l y h a l f t h e p h y s i c a l
length.
The l a y o u t a n d d i m e n s i o n s o f the c i r c u i r b o a r d o f
t h e t u n a b l e b a n d s t o p f i l t e r are s h o w n in Fig.
3.4.8.
The d e s i g n of the h o u s i n g for the t u n a b l e f i l t e r w a s
s l i g h t l y m o r e c o m p l e x than t h a t f o r the f i x e d f r e q u e n c y
f i l t e r s in t h a t d . c c o n n e c t o r s h a d to b e i n c o r p o r a t e d . A l s o
b o t h the h o u s i n g a n d the c i r c u i t b o a r d w e r e g o l d p l a t e d s o
t h a t g o o d o h m i c c o n t a c t s c o u l d be a c h i e v e d w h e r e d.c s h o r t
circuits were required.
P h o t o g r a p h n o 2 s h ows the a s s e m b l e d
f i l t e r w i t h th e t o p h a l f of the h o u s i n g removed.
The m e a s u r e d p e r f o r m a n c e o f the v a r a c t o r t u n e d b a n d ­
s t o p f i l t e r is s h o w n in Figs. 3 .4.9 to 3.4.11.
m a n c e is s u m m a r i s e d b e l o w
This p e r f o r ­
65
Bias Voltages
OV, 0.02V, 0 . 0 1 V
Centre Frequency
3.04 GHz
1 dB S t o p b a n d B a n d w i d t h
430 MHz
3 dB S t o p b a n d B a n d w i d t h
3 80 MHz
20dB Stopband Bandwidth
0 MHz
M i d - S t o p b a n d R e t u r n Loss
8 dB
P a s s b a n d R e t u r n Loss
> 20dB to 6 GHz
> 13dB to 7.2 GHz
Bias Voltages
4V, 4.41V,
Centre Frequency
3.602 GHz
1 dB S t o p b a n d B a n d w i d t h
421 MHz
3 dB S t o p b a n d B a n d w i d t h
310 M H z
20dB S t o p b a n d B a n d w i d t h
48 M H z
4.2V
xMid S t o p b a n d R e t u r n Loss
3.2 dB
P a s s b a n d R e t u r n loss
> 20 dB to 6.5 GHz
> 15 dB to 7.2 GHz
Bias Voltages
2 5.4V, 30V, 2 7.1V
Centre Frequency
4.21 GHz
1 dB S t o p b a n d B a n d w i d t h
417 M H z
3 dB S t o p b a n d B a n d w i d t h
207 M H z
20dB Stopband Bandwidth
62 MHz
M i d - S t o p b a n d R e t u r n Loss
1.7 dB
P a s s b a n d R e t u r n L oss
> 20 dB to 7 GHz
P h o t o g r a p h no.
2
Varactor Tuned Bandstop Filter
4-------------
Fig.
3.4.8
2 .11"
--------------»
L a y o u t a n d d i m e n s i o n s of the S.S.S ci r c u i t
b o a r d of the v a r a c t o r t uned b a n d s t o p filter
Fig.
3.4.9
tuned
-200MHz
Varactor
-30CMHZ
Bandstop
-lOCMKz
Filter
3 GHz
performance
+10CMHZ
at
zero
+20CMHZ
bias
+30CMHz
Fig. 3.4.10
Varactor
tuned
Bandstop
Filters
performance
with
bias
voltages
of
Fig.
3.4.11
Varactor
tuned
BandstoD
Filter
performance
with
bias
voltages
near
LO
lO
LO
LO
W
a s s e
rosi
30V
o
fN
66
T h e m e a s u r e d p e r f o r m a n c e of the f i l t e r w a s in c l o s e
a g r e e m e n t w i t h t h e o r y e x c e p t that the t u n i n g r a n g e w a s
s h i f t e d d o w n in f r e q u e n c y by a p p r o x i m a t e l y 400MHz.
This
w a s c a u s e d b y t he h i g h v a r a c t o r b o n d w i r e i n d u c t a n c e w h i c h
w a s n o t f u l l y a l l o w e d for in the design.
This could h o w ­
e v e r e a s i l y be c o m p e n s a t e d for by i n c r e a s i n g the v a l u e of
v a r a c t o r c a p a c i t a n c e u s e d in the d e s i g n process.
It is
i m p o r t a n t to n o t e t h a t the v a r a c t o r b o n d w i r e i n d u c t a n c e
li m i t s the m a x i m u m f r e q u e n c y of o p e r a t i o n .
This i n d u c t a n c e
w i l l r e s o n a t e the v a r a c t o r at 4.5 GHz at zero bias a n d 8.2G H z
at 60 Volts.
T h u s if a zero bias f r e q u e n c y of g r e a t e r th an
4.5 GHz is r e q u i r e d the v a r a c t o r m u s t be d e c o u p l e d to r e d u c e
its c a p a c i t a n c e as d e s c r i b e d in C h a p t e r 2.
In c h a p t e r 2 it w a s o b s e r v e d that m u l t i p l e v a r a c t o r
f i l t e r s c o u l d be t u n e d f r o m a s i n g l e p o w e r s u p p l y o v e r
t u n i n g b a n d w i d t h s of 10%.
In the a b o v e e x a m p l e , due to the
b r o a d e r t u n i n g b a n d w i d t h r equired,
i n d e p e n d e n t l y tuned.
the v a r a c t o r s h a d to be
T h i s p r o b l e m c o u l d be o v e r c o m e by c a r e ­
ful s e l e c t i o n of m a t c h e d v a r a c t o r s o r by u s e of a d i g i t a l
c o n t r o l s y s t e m w i t h th e r e q u i r e d v o l t a g e f r e q u e n c y c u r v e f o r
e a c h v a r a c t o r s t o r e d in s o l i d s t a t e memory.
3.5
T h e e f f e c t s of l a r g e R . F s i g n a l levels on v a r a c t o r s
tuned filter performance
In th e d e s i g n of v a r a c t o r t u n e d f i l ters the v a r a c t o r
is a s s u m e d to be a l i n e a r c i r c u i t e l e m e n t .
T h a t is, the
c u r r e n t t h r o u g h the d e v i c e is r e l a t e d to the v o l t a g e a c r o s s
the t e r m i n a l s of the d e v i c e by a l i n e a r d i f f e r e n t i a l equa t i o n .
67
In r e a l i t y ,
th e v a r a c t o r c a p a c i t a n c e is a f u n c t i o n of the
i n s t a n t a n e o u s v o l t a g e acro s s its t e r m i n a l s w h i c h is the
s u m of the d . c b i a s v o l t a g e and the i n s t a n t a n e o u s R.F s i g n a l
voltage.
T h u s t he v a r a c t o r c a p a c i t a n c e is a n o n - l i n e a r
c i r c u i t e l e m e n t w i t h the c u r r e n t and v o l t a g e r e l a t e d by
non-linear differential equations.
Parry
(3.6) has s h o w n
t h a t th e r e l a t i o n s h i p b e t w e e n the i n s t a n t a n e o u s c h a r g e s t o r e d
an d the v o l t a g e a c r o s s the t e r m i n a l s of an ideal a b r u p t
j u n c t i o n v a r a c t o r is g i v e n by the f o l l o w i n g e quation.
AV
4c 2 (VB) < v b + *>
(3.5.1)
(VB)
where
AV
Aq
VB
I n s t a n t a n e o u s v o l t a g e a c ross the t e r m i n a l s
=
Instantaneous charge stored
Bias Voltage
F r o m 3 . 5 . 1 it can be s e e n t h a t e v e n if the c h a r g e
f l o w i n g t h r o u g h the v a r a c t o r w e r e s i n u s o i d a l the t e r m i n a l
vol t a g e w o u l d possess second harmonic distortion.
In r e a l i t y
th e c h a r g e is n o t n e c e s s a r i l y s i n u s o i d a l and an i n f i n i t e
n u m b e r of h a r m o n i c s m a y b e gene r a t e d .
T h e l a rge s i g n a l
b e h a v i o u r of v a r a c t o r t u n e d f i l t e r s is n o t e a s i l y p r e d i c t e d
by analysis,
it is b e t t e r to p r e d i c t the b e h a v i o u r by a c t u a l
m e a s u r e m e n t s of p a r t i c u l a r filters.
T h e s e c o n d h a r m o n i c and f u n d a m e n t a l o u t p u t p o w e r s
of th e v a r a c t o r t u n e d b a n d s t o p f i l t e r w e r e m e a s u r e d as a
68
f u n c t i o n of i n p u t p o w e r and v a r a c t o r b i a s voltage.
These
w e r e m e a s u r e d by u s i n g a s p e c t r u m a n a l y s e r as the load
f o r the filter.
The m e a s u r e m e n t s w e r e t a k e n w i t h the
i n p u t s i g n a l at the same f r e q u e n c y as the s t o p b a n d c e n t r e
f r e q u e n c y of the filter.
Fig.
3 . 5 . 1 s hows the r atio of the s e c o n d and f i r s t
h a r m o n i c p o w e r s as a f u n c t i o n of i n p u t p o w e r at a bias
v o l t a g e of a p p r o x i m a t e l y - I V on each vara c t o r .
E x a m i n i n g Fig.
3.5.1 w e see that at +5 d B m i n p u t
p o w e r the s e c o n d h a r m o n i c w a s 17 dB down on the f u n d a m e n t a l
out p u t .
N o e q u i p m e n t w a s a v a i l a b l e for s u p p l y i n g m o r e t h a n
5 d B m of power, h o w e v e r the c u r v e h a s a p p r o x i m a t e l y l i n e a r
s l o p e an d b y l i n e a r e x t r a p o l a t i o n w e see t h a t at +13 d B m
i n p u t p o w e r the s e c o n d h a r m o n i c p o w e r is e qual to that of
the f u n d a m e n t a l .
T h i s is k n o w n as the s e c o n d o r d e r e d i n t e r ­
c e p t point.
M e a s u r e m e n t s t a k e n as a f u n c t i o n of b i a s v o l t a g e
s h o w e d a g e n e r a l 6 d B m i n c r e a s e of i n t e r c e p t p o i n t e a c h t i m e
the b i a s v o l t a g e w a s d oubled.
U n f o r t u n a t e l y w h e n the
h a r m o n i c d i s t o r t i o n w a s b e l o w -45 d B m it was d i f f i c u l t to
m e a s u r e a b o v e the s y s t e m noise, b u t by e x t r a p o l a t i o n the
i n t e r c e p t p o i n t at 10 V bias w o u l d be 38dBm.
T h e p h y s i c a l b a s i s for the r e d u c t i o n of d i s t o r t i o n
w i t h i n c r e a s e d b i a s v o l t a g e is that the v a r a c t o r c a p a c i t a n c e
b e c o m e s less s e n s i t i v e t o b i a s as the bias is i n c reased.
No third or higher ordered harmonics were detected.
T h e p h y s i c a l b a s i s for this is that the r e s o n a t o r s are o p e n
c i r c u i t e d at f r e q u e n c i e s a w a y f r o m the s t o p b a n d c e n t r e
Pin
Fig.
3.5.1
(d B m )
R a t i o of S e c o n d h a r m o n i c to F u n d a m e n t a l
o u t p u t p o w e r s as a f u n c t i o n of i n p u t
p o w e r for the v a r a c t o r t u n e d b a n d s t o p
filter
69
f r e q u e n c y a n d thus no h a r m o n i c curr e n t s can flow in the
resonators,
t h e r e f o r e in 3 .5.1 only s e c o n d h a r m o n i c v o l t a g e s
w i l l be g e n e r a t e d .
3.6
Conclusions
A n o v e l d e s i g n t e c h n i q u e for s y m m e t r i c a l b a n d s t o p
f i l t e r s has b e e n d e v e l o p e d a n d e x p l i c i t d e s i g n f o r m u l a e are
presented.
T h i s t h e o r y has b e e n u s e d in the d e s i g n of S . S . S
f i x e d f r e q u e n c y fi l t e r s and v a r a c t o r t u n e d filters.
Computer
loss a n a l y s i s of the f i l t e r s is p r e s e n t e d a n d this a g rees
c l o s e l y w i t h the m e a s u r e d p e r f o r m a n c e s of e x p e r i m e n t a l devices.
References
3.1
'Theory of E l e c t r i c a l Filt e r s '
pp70-71
J.D. R h o d e s
Wiley
3.2
'Advanced Engineering Mathematics'
Erwin Kreysig
Wiley
3.3
'Microwave Filters,
Impedance Matching Networks
and Coupling Structures'
Matthaei, Young and Jones
M c G r a w Hill
3.4
R e f 3.3
Chapter 2
70
3. 5
Ref
3.1 p p 4 8 - 4 9
3.6
' M i c r o w a v e F r e q u e n c y Divi d e r s '
R. P a r r y
I n t e r n a l R e p o r t for the D e g r e e of Ph.D
D e p a r t m e n t of E l e c t r i c a l E n g i n e e r i n g ,
T h e U n i v e r s i t y of L e eds
71
4.
TUNABLE MICROWAVE BANDPASS FILTERS
4.1
Introduction
I n i t i a l a t t e m p t s to c o n s t r u c t a t u n a b l e b a n d p a s s
f i l t e r w e r e d e s c r i b e d in c h a p t e r two.
This f i l t e r s u f f e r e d
f r o m a f r e q u e n c y r e s p o n s e shape w h i c h w a s h i g h l y d e p e n d e n t
on tuned frequency.
This p r o b l e m w a s a c o n s e q u e n c e o f the
f r e q u e n c y d e p e n d e n t c o u p l i n g b e t w e e n the r e s o n a t o r s o f the
f i l t e r , th e f i l t e r w a s thus m i s m a t c h e d w h e n it w a s t u n e d
a w a y f r o m the d e s i g n e d c e n t r e freq u e n c y .
To o v e r c o m e this p r o b l e m , a n o v e l t u n a b l e c o m b l i n e
f i l t e r ha s b e e n d e v e l o p e d .
By s u i t a b l e d e s i g n of the
r e d u n d a n t t r a n s f o r m e r e l e m e n t s at the i n p u t a n d o u t p u t of
t h e c o m b l i n e f ilter,
it is p o s s i b l e to a p p r o x i m a t e l y c a n c e l
o ut th e f r e q u e n c y d e p e n d e n c e of the c o u p l i n g b e t w e e n the
r e s o n a t o r s so t h a t a g o o d r e s p o n s e can b e m a i n t a i n e d o v e r
large tuning bandwidths.
The f i l t e r a l s o p o s s e s s e s the
additional important property of maintaining approximately
c o n s t a n t a b s o l u t e b a n d w i d t h o v e r v e r y l a rge t u n i n g ranges.
D e s i g n p r o c e d u r e s are also p r e s e n t e d for a t u n a b l e
c o m b l i n e f i l t e r s u i t a b l e for v e r y n a r r o w p a s s b a n d a p p l i c a t ­
ions .
This d e v i c e u s e s o p e n - c i r c u i t e d r e s o n a t o r s thus a v o i d ­
i n g l o s s y s h o r t c i r c u i t s w h i c h can b e c o m e a p r o b l e m w h e n
p a s s b a n d b a n d w i d t h s o f less than 5% are required.
T h e d e s i g n of a p u r e l y d i s t r i b u t e d c i r c u i t s u i t a b l e
f o r t h e r e a l i s a t i o n of n a r r o w b a n d S . S . S b a n d p a s s f i l t e r s is
also presented.
72
The p r a c t i c a l d e s i g n a n d m e a s u r e d p e r f o r m a n c e s of
f o u r f i l t e r s are p r e s e n t e d .
The first is a two r e s o n a t o r
v a r a c t o r t u n e d c o m b l i n e f i l t e r d e s i g n e d to tune f r o m 3GHz
to 5GHz w i t h a p a s s b a n d b a n d w i d t h o f 5%.
T h e s e c o n d is a
3% b a n d w i d t h t hree r e s o n a t o r v a r a c t o r t u n e d o p e n c i r c u i t e d
c o m b l i n e f i l t e r d e s i g n e d to tune f r o m 3.5G H z to 4.5GHz.
The o t h e r d e v i c e s w e r e f i x e d f r e q u e n c y 4GHz three r e s o n a t o r
f i l t e r s w i t h 1% a n d 5% p a s s b a n d b a n d w i d t h s .
exhibited measured performances
All four filters
in e x c e l l e n t a g r e e m e n t w i t h
theoretical expectations.
4.2
T h e o r e t i c a l D e s i g n of T u n a b l e M i c r o w a v e B a n d p a s s F i l t e r s
4 . 2 . 1 T h e o r y of T u n a b l e C o m b l i n e F i t l e r s
T h e c o m b l i n e f i l t e r is s h o w n in Fig.
4.2.1.
It c o n s i s t s
o f a c o m m e n s u r a t e m u l t i - w i r e line [ 4 . 1 ] . w i t h c o u p l i n g
c o n s t r a i n e d to b e b e t w e e n a d j a c e n t lines.
T h e l i n e s are
e a c h s h o r t c i r c u i t e d to g r o u n d at the same e n d w h i l s t o p p o s i t e
e n d s ar e t e r m i n a t e d in l u m p e d c a p a c i t o r s .
f r e q u e n c y o f th e fil t er,
At the r e s o n a n t
the l i n e s are s i g n i f i c a n t l y less
t h a n on e q u a r t e r o f a w a v e l e n g t h long, i n d e e d if t h e l u m p e d
c a p a c i t o r s w e r e n o t p r e s e n t a n d the lines w e r e o n e q u a r t e r
w a v e l e n g t h l o n g at r e s o n a n c e the n e t w o r k w o u l d b e an all
stop structure.
T w o a d v a n t a g e s o f this f i l t e r a r e its c o m p a c t
s i z e a n d v e r y b r o a d s t o p b a n d b a n d w i d t h , b o t h o f w h i c h hav e
c o n t r i b u t e d t o its u s e in m u l t i p l e x e r d e s i g n [4.2],
The e q u i v a l e n t c i r c u i t of the c o m b l i n e f i l t e r is s h o w n
in Fig.
4.2.2.
In Fig.
4 .2.2 the r e s o n a t o r s are c o m p o s e d o f
d i s t r i b u t e d i n d u c t o r s in p a r a l l e l w i t h l u m p e d c a p a c i t o r s ,
w h i c h the c o u p l i n g b e t w e e n r e s o n a t o r s is a c h i e v e d b y s e r i e s
distributed inductors.
r
r
Fig.
1
4.2.1
The Combline Filter
1
Y
.
n-l,n
Fig.
4.2.2
E q u i v a l e n t c i r c u i t of the C o m b l i n e F i l t e r
The m e t h o d of d e s i g n i n g the c o m b l i n e f i l t e r c o n s i s t s
of t r a n s f o r m i n g the e q u i v a l e n t c i r c u i t into a n e t w o r k
c o m p o s e d of s h u n t r e s o n a t o r s c o u p l e d b y a d m i t t a n c e i n v erters.
The i n v e r t e r s are f o r m e d by s h u n t i n g the s e r i e s c o u p l i n g
i n d u c t o r s by i d e n t i c a l e l e m e n t s o f o p p o s i t e sign [Fig.
4.2.3],
The t r a n s f e r m a t r i x b e t w e e n the rth a n d r + i t h n o d e s in the
n e t w o r k is t h e n as f o l l o w s
n
1
7
jtan (aw)
1
0
jYr,r+l
tan (aw)
1
Y r,r+1
j Y r,r+l
tan (aw)
0
1
0
1
jtan(au))
Yr,r+1
(4.2.2)
yy r,r-KL
tan (aw)
w h i c h is the t r a n s f e r m a t r i x o f an a d m i t t a n c e i n v e r t e r
of c h a r a c t e r i s t i c a d m i t t a n c e K
K
r,r+l
F r o m Fig.
=
Y
,, w h e r e
r ,r+l
1/tan(aw)
r ,r+l
(4.2.3)
4.2.3 the rth r e s o n a t o r c a n b e seen to h a v e
an a d m i t t a n c e Y(r) w h e r e
Y +
r
f ,
r-l,r
+Y
'r+1
K
r,r+l
Y
'r+1
r+1
+Y
+ Y
r,r+1
r+1,r+2
r,r+l
Fig.
4 .2.3
F o r m a t i o n of a d m i t t a n c e i n v e r t e r s in the
combline filter
74
Y(r) =
j[u)Cr - (Yr +
+ Yr ^ +1)/tan(ao))]
(4.2.4)
The f r e q u e n c y d e p e n d e n c e of the a d m i t t a n c e i n v e r t e r s
is n o w r e m o v e d by s c a l i n g the e n t i r e n e t w o r k a d m i t t a n c e ,
i n c l u d i n g the t e r m i n a t i n g r e s i s t o r s by a f a c t o r tan(aco)/
tan(acuo).
After scaling we obtain
K r,r+ 1
"
Y r , r + l / t a n < a “o>
(4-2 '5 >
an d
Y(r) '
tSnLp
w h e r e u>
C“cr tan(ao,) - j^r + V l , r + Yr,r+1)3
<4'2 -6)
is the p a s s b a n d c e n t r e f r e q u e n c y of the filter.
F r o m the l o w p a s s p r o t o t y p e f i l t e r s h o w n in Fig.
4.2.4
w e o b t a i n the l o w p a s s to b a n d p a s s f r e q u e n c y t r a n s f o r m a t i o n
be low
to
a[ Bwtan (aw) - 1 ]
(4.2.7)
where
“ = [ Y r + Y r - l , r + Y r , r + l ] / [ C L r t a n < a “o ' ]
(4‘2 -8)
B = C /[Y
(4.2.9)
and
r
r
+ Y
.
+ Y
.n ]
r-l,r
r ,r+l
E q u a t i n g u> = ±1
in the lowp a s s p r o t o t y p e to the b a n d -
e d g e s oil a n d co2 in th e c o m b l i n e f i l t e r w e o b t a i n
•
1
1
1
K12
T
,C> T
1
K ,
n-l,n
T -
T,
c>
1
Kr,r+!
“
- r cn
---- 1-----O
Fig. 4.2.4
The Lowpass Prototype Filter
Fig.
T h e t r a n s f o r m e r e l e m e n t s at the i n p u t
4.2.5
an d o u t p u t of the c o m b l i n e f i l t e r
I
'■**
75
-1
=
a(go)1 tanfaa^) - 1 )
(4.2.10)
+1
=
ct(Bu)2 t a n ( a u 2 ) - 1 )
(4.2.11)
The b a n d c e n t r e f r e q u e n c y
caQ is d e f i n e d as the r e s o n a n t
f r e q u e n c y o f the f ilter, thus
g(jjQ tan (au)o ) - 1
=
0
(4.2.12)
or
3
=
1 / (uQ tan (an)Q ) )
(4.2.13)
F o r n a r r o w p a s s b a n d b a n d w i d t h s Aw we c a n a p p r o x i m a t e
t h e b a n d e d g e f r e q u e n c i e s as follows
0)^
=
w o - Aa)/2
(4.2.14)
u)~
=
co
+ A co/2
(4.2.15)
A cj
< <
2.
O
with
a)
o
U s i n g 4.2. 1 4 a n d the t r i g o n o m e t r i c e x p a n s i o n o f tan(A+B) w e
obtain
tan (aiu ) - a Aw/2
tan(aw,)
1
Now
2
“
■
*— — . °,~ ,-------- r
1 + a Aw/2 tan (am
)
(4.2.16)
t a n ( a w ) << 1
o
Thus e x p a n d i n g [ 4 . 2.16] u s i n g t h e b i n o m i a l t h e o r u m w e o b t a i n
76
tan(aco1 ) « [ t a n f a ^ )
- ^
] [1 -
tan (aa> ) ]
(4.2.17)
tan(aw o )]
(4.2.18)
Similarly
tan (aco0
) » [tan(aw o ) + ^ z] [ 1
Z
+ ^z
N o w d e n o t i n g ao)o as 0q , a n d s u b s t i t u t i n g 4.2.17 into 4 . 2 . 1 0
a n d 4 . 2 . 1 8 i n t o 4.2.11, a n d i g n o r i n g s e c o n d o r d e r t erms in
Aw, t h e n
-1 = a [B[ai ta n (0 ) - 4^ [tan(e ) + 0 [ l + t a n 2 (0 )]]]-l]
o
o
z
o
o
o
(4.2.19)
And
+1 = aCB[w tan(0 ) + 4^ [tan(0 ) + 0 [l + tan2 (0 )]]]-l]
0
0
2
o
o
o
(4.2.20)
S o l v i n g 4 . 2 . 1 9 a n d 4 . 2 . 2 0 s i m u l t a n e o u s l y f o r Aw y i e l d s
A
=
---------------- 1---------- -------a 6 [tan (0 ) + e ( l + t a n 2 (0 ))]
(4.2.21)
F r o m 4 . 2 . 1 3 a n d 4 . 2 . 2 1 w e o b t a i n the e x p r e s s i o n for Aw as
a f u n c t i o n of t u n e d f r e q u e n c y
Au>
=
2 as tan(0 )
---------------2------ ° ------a [ t a n ( 9 o ) + 0Q (l+tan (0Q ))]
(4.2.22)
T he f r e q u e n c y d e p e n d e n t p a r t o f 4 . 2.22 is F ( 0 q ) w h e r e
77
6 t a n (0 )
----------- ^------ 2----- 2-----tan (0 ) + 6 [ 1 + tan (0 ) ]
o
o
o
F(6 ) =
For 0
o
(4.2.23)
small we have
t a n (0 )
o
=0
(4.2.24)
o
and
9
0
F(6o )
“ 2~6~ —
o
(4.2.25)
w h i c h is an i n c r e a s i n g f u n c t i o n
As 0
o
t e n d s to 90° w e h a v e
tan (6 )
-
(4.2.26)
and
F(6 )
o
->0
(4.2.27)
T h u s the b a n d w i d t h o f t h e c o m b l i n e f i l t e r h a s a t u r n i n g p o i n t
somewhere between 6
= 0 ° a n d 0 = 90°.
o
o
I n v e r t i n g 4.2.2 3 w e o b t a i n
,
o
Differentiating
,
0O
1 + t a n 2 (0 )
[F (0 ) ] 1 =
tan(0Q )
0^
o
sin(20 o )
(4.2.29)
-± -
+
—
---(4.2.28)
a n d e q u a t i n g to z e r o to e v a l u a t e
the turning point we obtain
78
— 2
0
o
4 c o s (20 )
----2-----~
sin (20 )
o
+
=
0
(4.2.30)
4 . 2 . 3 0 c a n be s o l v e d n u m e r i c a l l y u s i n g the N e w t o n R a p h s o n
t e c h n i q u e to o b t a i n
0q
=
52.885°
(4.2.31)
T h i s v a l u e o f 0q is the p o i n t at w h i c h the b a n d w i d t h
o f t h e c o m b l i n e f i l t e r is m a x i m i s e d .
T h e f u n c t i o n F(0 ) is
o
s h o w n b e l o w a n d it ca n b e s e e n that t h e f i l t e r b a n d w i d t h is
approximately constant,
i n d e p e n d e n t of t u n e d freq u e n c y , e v e n
over octave tuning bandwidths.
The filter should be designed
so t h a t the e l e c t r i c a l p h a s e l e n g t h s of t h e r e s o n a t o r s are
52.885° long at th e e x a c t c e n t r e of t h e t u n i n g band.
F(0o )
0.237
O
30°
0.288
50°
0.314
60°
0. 306
o
o
f"
0
o
CN
0.167
0
0
O
0.254
W i t h the f i l t e r d e s i g n e d in t h i s m a n n e r , o c t a v e
t u n i n g a r o u n d eQ = 52.885° w i l l c a u s e less t h a n a 20% c h a n g e
in a b s o l u t e b a n d w i d t h .
The scaling of the network admittance performed
p r e v i o u s l y r e s u l t e d in f r e q u e n c y d e p e n d e n t t e r m i n a t i n g
r e s i s t o r s of c o n d u c t a n c e
79
GL
=
GS
=
tan (au)/tan (aa>o )
(4.2.32)
T u n i n g the f i l t e r o v e r an o c t a v e w i t h 0 q = 52.9° at
m i d b a n d w i l l c h a n g e the t e r m i n a t i n g r e s i s t o r s b y a f a c t o r
of 3.92:1.
S u c h a v a r i a t i o n is u n a c c e p t a b l e b e c a u s e it w o u l d
r e s u l t in a v e r y p o o r f i l t e r r e s p o n s e at t u n e d f r e q u e n c i e s
a w a y f r o m the o p t i m u m value.
In o r d e r to m a i n t a i n a g o o d
r e s p o n s e o v e r b r o a d t u n i n g b a n d w i d t h s this f r e q u e n c y v a r i a t i o n
mus t be removed.
Redundant, non-resonated transformer stubs
ar e i n t r o d u c e d at th e i n p u t a n d o u t p u t of the f i l t e r to a c h i e v e
t h i s o b j e c t i v e [Fig.
4.2.5],
It is p r o v e n in A p p e n d i x 4.1
t h a t the v a l u e s of the a d m i t t a n c e s of t h e t r a n s f o r m e r s tubs
must satisfy
YQ
=
1 - 1 / c o s (6 )
(4.2.33)
Y_, =
Ol
1 / c o s (0 )
o
(4.2.34)
Y ' =
1
1 - 1 / c o s (0 )
o
(4.2.35)
W i t h t h e s e v a l u e s of t r a n s f o r m e r e l e m e n t s the
a d m i t t a n c e of the l o a d s e e n b y the f i l t e r l o o k i n g b a c k into
Yj1 / a n d a f t e r s c a l i n g b y tan (acu) / t a n (au>Q ) is
Y
=
L
s i n (26)
sin(2e )
o
(cos(26) - cos(26o ))
s i n (26 5
o
is r e s o n a n t at 0 q a n d t h e r e q u i r e d m i n i m u m v a r i a t i o n in
t h e r e a l p a r t of Y T is a c h i e v e d .
L
W i t h 0 = 45° at t h e m i d d l e
o
o f t he t u n i n g b a n d , t h e n o c t a v e t u n i n g f r o m 30° to 60° c h a n g e s
t h e l o a d a d m i t t a n c e f r o m 0 . 8 6 6 to 1, thus o n l y
a very small
80
d e t e r i o r a t i o n in f i l t e r r e s p o n s e w i l l b e noticed.
T o a c h i e v e r e a l i s a b l e e l e m e n t values,
the a d m i t t a n c e
l e v e l of the f i l t e r m u s t b e s c a l e d at each n o d e in o r d e r
t h a t th e h i g h a d m i t t a n c e s of the s h u n t r e s o n a t o r s can be
a b s o r b e d i n t o the c o u p l i n g e lements.
Scaling also allows
t h e l u m p e d c a p a c i t o r s to h a v e any d e s i r e d e q u a l v a l u e w h i c h
is i m p o r t a n t for the r e a l i s a t i o n of p r a c t i c a l v a r a c t o r t u n e d
filters.
A f t e r s c a l i n g the rth n o d e in the f i l t e r b y a f a c t o r
2
n ^ th e t y p i c a l c o u p l i n g n e t w o r k b e t w e e n n o d e s r a n d r+l w i l l
b e as s h o w n in Fig.
4.2.6.
After absorbing the ideal trans­
f o r m e r s in th e c o u p l i n g n e t w o r k into the i n verters, w e o b t a i n
K r , r + l = n r .n r + l Y r , r + ln'/ tan (e o )
(4.2.37)
The s c a l i n g c o n s t a n t a b e c o m e s
_
n 2 [Y + Y
.
+ Y
r , r+l
r
r_____ r - l , r
CL tan ( 0 5
r
o
(4.2.38)
Th e i d e a l t r a n s f o r m e r s w h i c h r e m a i n at the i n p u t a n d
o u t p u t o f the f i l t e r [Fig.
4.2.7] c a n n o t b e a b s o r b e d into
t h e i n v e r t e r s a n d m u s t b e a b s o r b e d i n t o the r e d u n d a n t t r a n s ­
f o r m e r stubs.
It can b e s h o w n b y a p r o c e d u r e s i m i l a r to
t h a t in A p p e n d i x 4.1 t h a t the v a l u e s of t h e t r a n s f o r m e r
e l e m e n t s w i l l b e m o d i f i e d as b e l o w
Y
1 - 1/n, cos ( 0 )
X
o
(4.2.39)
YQ1 =
1 / n ^ c o s (0Q )
(4.2.40)
Y1
1 /Tll
(4.2.41)
O
=
=
" 1 / n 1 COS ( 6o )
nr : 1
Fig.
4.2.6
1 : nr
nr+1 : 1
1 : r]r+1
S c a l i n g p r o c e d u r e on i n t e r n a l nodes of the
combline filter
Y(l)
Fig.
4.2.7
T he i n p u t
(output) of the c o m b l i n e filter
a f t e r s c a l i n g of the i n t e r n a l nodes
81
T hese v a l u e s w i l l y i e l d the r e q u i r e d load frequency
v a r i a t i o n g i v e n in 4.2.36 a n d the c o n d i t i o n for realisability is n o w t h a t Y q is p o s i t i v e .
n ^ £ l / c o s ( 0 o ).
This w i l l b e true if
T y p i c a l l y , p a s s b a n d b a n d w i d t h s of less
t h a n 30% w i l l a l l o w this c o n d i t i o n to b e satisfied.
E x a m i n i n g 4.2 . 36 w e see t h a t the load admi t t a n c e
ha s a r e s o n a n t i m a g i n a r y part, w h e n the filter is t uned to
f r e q u e n c i e s at w h i c h this is n o t zero this a d m i t t a n c e w i l l
a f f e c t the f i r s t a n d last res o n a t o r s .
To c o m p e n s a t e for
t h i s the f i r s t a n d last r e s o n a t o r s w i l l be s c a l e d as follows.
F r o m 4 . 2 . 7 a n d 4.2.36 the a d m i t t a n c e of t h e first
(and last)
r e s o n a t o r is
Y(l)
=
C L ^ a ( B w tan (au)-l)
+
cos(2aw) - cos(2aw )
--------- r-7^---- r---- SL.
sin 2aoio )
(4.2.42)
T h e r e q u i r e d v a l u e of this r e s o n a t o r a d m i t t a n c e is
Y (1)
=
C L ^ a [ 3 to tan ( a w ) -1]
(4.2.43)
T h e v a l u e o f a in 4 . 2 . 4 2 w i l l n o w b e m o d i f i e d so
t h a t 4 . 2 . 4 2 w i l l p o s e s s the same s u s c e p t a n c e s l ope p a r a m e t e r
as 4 . 2 . 4 3 a t w^.
Thus, d i f f e r e n t i a t i n g 4 . 2 . 3 0 a n d 4.2.31
a n d e q u a t i n g a t a)Q w i t h the m o d i f i e d v a l u e o f a in 4.2.42
b e i n g a' w e o b t a i n
2a = C L x (a ' - a)[B tan ( 0Q ) + B 0 Q C 1 + t a n 2 (0q ) ]]
(4.2.44)
82
Re arr anging we obtain
a1 = a
+
2a
C L ^ g [t a n (6q ) + 0Q ( 1 + t a n 2 (0Q ) )]
(4.2.45)
F r o m 4 . 2 . 4 5 a n d 4.2 . 2 1
(4.2.46)
w h i c h is th e r e q u i r e d s c a l i n g f a c t o r for r=l a n d r=n.
T h e d e s i g n e q u a t i o n s for t h e t u n a b l e c o m b l i n e f i l t e r
can n o w b e e v a l u a t e d .
F r o m 4.2.9 and 4.2.13 we obtain
[ Y r + Y r - r , r + Y r , r + 1 ] = C “o tan < V
( 4 '2 -4
w h e r e C is t h e v a l u e of the l u m p e d t u n i n g c a p a c i t o r at
b a n d centre.
F r o m 4 . 2 . 2 2 a n d 4.2.4 6
2 id tan ( 0 )
a
(4.2.48)
A oj[ ( 0Q ) + 0 Q ( 1 + t a n 2 (0 ) ) ]
[f o r
r = 2
to n - 1]
and
a
[ f o r r = 1 a n d r = n]
From 4.2.38
(4.2.49)
83
n
r
Fr o m 4.2.37
Y
(4.2.51)
r ,r+l
Finally from 4.2.47 and 4.2.51
can b e o b tained.
T h u s in s u m m a r y, w e i n i t i a l l y o b t a i n the v a l u e s C
J_jK
and
fo r th e l o w p a s s p r o t o t y p e filter, this w o u l d
/
n o r m a l l y b e t h e C h e b y s h e v p r o t o t y p e f i l t e r [4.3].
N e x t the
v a l u e o f 0 q at the m i d d l e of the t u n i n g b a n d m u s t b e chos e n ,
f o r o p t i m u m p a s s b a n d r e t u r n loss a c r o s s t h e t u n i n g b a n d 0Q
s h o u l d b e 45°, if c o n s t a n t b a n d w i d t h is r e q u i r e d 0 q s h o u l d
b e 52.9°.
T h e v a l u e o f the l u m p e d t u n i n g c a p a c i t o r s h o u l d
t h e n b e c h o s e n a n d th e a d m i t t a n c e o f the c o m b l i n e f i l t e r can
t h e n b e d e r i v e d b y a p p l y i n g e q u a t i o n s 4 . 2 . 4 7 to 4 . 2 . 5 1 a n d
4.2.39
to 4.2.41.
For a Chebyshev prototype filter only
th e f i r s t h a l f of t h e e l e m e n t v a l u e s n e e d to b e e v a l u a t e d
s i n c e the n e t w o r k w i l l be s y m m e t r i c a l .
4.2.2
T h e o r y of T u n a b l e O p e n - C i r c u i t e d C o m b l i n e F i l t e r s
L o s s y s h o r t c i r c u i t s can b e c o m e a p r o b l e m in the
p r a c t i c a l r e a l i s a t i o n of S . S . S c o m b l i n e f i l t e r s w i t h n a r r o w
passband bandwidths.
A l s o in the c a s e o f v a r a c t o r t u n e d
c o m b l i n e f i l t e r s , s i n c e the v a r a c t o r s are h e a v i l y c o u p l e d into
the resonators,
certain requirements may not be met because
o f e n e r g y d i s s i p a t i o n in the varactors.
84
An alternative to the tunable combline filter is
the tunable open-circuited combline filter in which the
resonators are between one quarter and one half wavelength
long at b a n d c e n t r e , and no short circuits are required.
The varactors are not as heavily coupled into the resonators
of this filter,
since, because the resonators are longer,
they contain more stored energy than the normal combline
filter.
Consequently for a given resonant frequency and
p a s s b a n d bandwidth this filter will have a lower dissipation
loss than the tunable combline filter.
In Appendix 4.2 it
is shown that the Q factor of this filter will be approxi­
m a tel y twice that of the tunable combline filter.
ately,
Unfor tu n­
the inevitable trade off is that of tuning bandwidth.
The tunable combline filter will tune from 30° to 60° with a
6:1 capacitance variation,
the open-circuited combline filter
wil l only tune from 120° to 150° with the same capacitance
ratio.
The ba sic physical structure of this filter is shown
in Fig.
4.2.8.
The structure is essentially the same as the
normal combline filter except that the short circuits have
been removed and replaced by open circuits.
The equivalent
circuit of the filter is identical to that shown in Fig.
4.2.2
ex cept that the inductor symbols now represent open circuited
stubs with electrical length of between one quarter and one
half wa vel eng th at b and centre.
The admittance of the rth resonator in the open
. circuited combline filter is
Y(r)
=
uC + j Y r tan (aw)
4.2.52
lumped tuning
capacitor
X
1
open circuited
stubs
Fig.
4.2.8
The Open Circuited Combline Filter
1
85
Comparing this admittance with the admittance of the c om b ­
line resonator it can be seen that it is identical e xc ep t
that tan(aw) has been replaced by -1/tan(aw).
Using this
substitution throughout section 3.2.1 results in the design
equations for this filter.
For this reason the proof of
the design is not presented here,
the equations are merely
stated below.
The Design Process
Firstly compute the element values CL^ and K^_
f
in the lowpass prototype filter.
Choose C, the lumped tuning capacitor at ba n d centre.
Choose the mid ban d phase lengths of the resonators,
this should be 135° if optimum return loss is required and
136.26° if constant b andwidth is required,
there is however
no significant difference between these two values.
N e x t calculate the admittance to ground at each node
in the filter from
(4.2.53)
Calculate the scaling factor of each resonator from
a
t a n (0 ) A w [ t a n (6 )-9 (1+tan
o
o
o
2
(4.2.54)
(0 ))]
o
[for r = 2 to n - 1 ]
And
(4.2.55)
[for r = 1 and r = n]
86
Calculate the scaling factor at each node from
CL a
nr
-
C (Y
r
+ Y
"
r-l,r
+ ~Y
r,r-t-l
)t a n (0 ) ]
o
(4.2.56)
Calculate the coupling admittance between the stubs
from
Y r ,r +1 = 'K r,r+l/ C n r-n r+l tan (V
]
(4.2.57)
Calculate the admittance to ground of the rth stub
from
Y r = -C„o/tan(eo ) - y r . 1/r - Y r ,r+1
(4.2.58)
Finally calculate the admittances of the transformer
elements
from
Y
4.2.3
=
1 - l/n]sin(0
)
(4.2.59)
Y , =
ol 1
l / n 1 sin (0 )
(4.2.60)
Y 1I =
1 /n ^2 - l/n 1 s i n ( 0Q )
(4.2.61)
o
Theory of the Distributed Combline Filter
The theory of tunable combline filters is now extended
to the design of a distributed combline filter suitable for
S.S.S
realisation of narrow b a n d fixed frequency bandpass
filters.
Although either of the previous two filters can b e
used as fixed frequency filters they both require lumped
capacitors in order for them to operate.
This w o u l d not at
first seem a prob lem since the capacitors could be realised
using tuning screws.
However,
low loss filters require large
87
g r o u n d plane spacings and the screws must then penetrate
deep into the cavity.
The inductance of the screws can
then seriously deteriorate the filter performance.
It
w o u l d thus be desirable if a design procedure could be
de v e l o p e d for fixed frequency filters requiring no lumped
capacitors.
The advantages of using a design technique base d on
the tunable combline filter are two fold.
Firstly the
re dundant transformer elements cancel the frequency variation
of the coupling bet wee n the resonators,
passbands of up to 30% can be achieved.
thus low V.S.W.R.
Secondly, because
the filter w il l inevitably require fine tuning,
the advantage
of h a v i n g approximately constant bandwidth independent of
tuned frequency is obvious.
The S.S.S circuit configuration of the distributed
combline filter is shown in Fig.
4.2.9.
This filter is
relatively easy to analyse if the input and output feed
terminals are
located along the dotted lines in Fig.
In that case,
the e qui valent circuit of the filter is as shown
in Fig.
4.2.10.
The inductor symbols in Fig.
the long sections shown in Fig.
4.2.9.
4.2.10 represent
4.2.9, these b ein g open
circuited stubs of between one quarter and one hal f wavelength
long
at b a n d centre.
sections,
The capacitors represent the short
these being open circuited stubs which are less than
one quarter wavelength long.
With coupling allowed only
between adjacent short sections it is obvious that the equivalent
circuit of the filter is similar to that of a combline filter
Ccmpcnents of open circuited n
wire line
\
0/C stub
greater than A/4 in length
Fig.
Fig.
4.2.9
4.2.10
The distributed combline filter structure
Equivalent circuit of the distributed combline
filter
88
with
the n wire line composed of the short stubs and the
r e sonating elements being the distributed inductive openc ircuited stubs.
In Fig.
inductive stubs,
4.2.10 Yb represents the admittance of the
these all being identical.
The propogation
constants of the short and long stubs are denoted
'a' and
'b' respectively.
The process of obtaining a set of design equations
for this filter is very similar to that for the previous two
filters.
Firstly,
admittance inverters are formed,
then
the frequency dependence of the inverters is scaled into the
terminating resistors and the network is scaled at each node
to obtain realisable element values.
The frequency trans­
formation is
cj
-*■ a [ 3 tan (b w) / tan (am) + 1 ]
(4.2.62)
where
+ Y
r,r+l
]
(4.2.63)
A nd
(4.2.64)
tan
an (aw )
o
tan(bu) )
o
For narrow bandwidths the passband bandwidth of
• the filter is approximated by
89
si n ( 2 atoo ) sin (2bw
a [a sin (2b
~
oj
o
)
)-bsin(2aoo
o
)]
We require a turning point of 4.2.65,
-
°
(4.2.65]
thus
U = „
1
o
(«•*■««>
or
2
a cos (2ato )
---- 2---- -sin (2 au) )
o
=
2
b cos (2bu )
?------ —
sin (2bo> )
o
(4.2.67)
The most obvious solution of 4.2.6 7 is that aw
o
=4 5°
Q
and bu)Q = 135 .
Unfortunately, with this solution,
from
4.2.64 we ob tain
lx
r
-r i
r-l,r
+ Y
,]
r,r+l
(4.2.68)
This is an unacceptable solution because for a
finite physical separation between the long stubs we require
Y^ greater than Y b .
Solutions of 4.2.67 must thus be
constra in ed to the following set
au
< 45°
o
(4.2.69)
bu)
o
> 135°
For example aojQ = 30° and b w o = 136° are solutions
of 4.2.67 and from 4.2.64 we then have
Yb
=
0 . 5 7 [ Y r + Y r_ ljr + V r > r + 1 :
(4.2.70)
For narrow bandwidths Y r-1 r and Y r r+1 will be
small and we thus have
90
Yb
«
0.57 Y r
(4.2.71)
This implies a finite physical separation between
the long stubs.
The Design Process for the Distributed Combline Filter
As stated,
the design process of this filter is
very similar to that for the tunable combline filter.
design process
The
is n ot derived in detail, merely summarised
below.
Firstly obtain the lowpass prototype element values
K
. and CL .
r,r+l
r
Choose a<Jo and b u 0 to satisfy 4.2.67 under the
constraint of 4.2.69
Choose Yb
Calculate the admittance to ground at each node in
the filter from
[Y
r
+ Y
.
+ Y
r-l,r
r,r+l
Calculate
- Yb tan (bw )
— r---7--- r2tan(acoo )
=
(4.2.72)
the resonator scaling factor a from
s i n ( 2aw ) s i n ( 2b w
)
________
ct = —
--- :— o__________ o_________
A w[ a sin (2bw
o
) -b sin (2aw ) ]
o
(4.2.73)
[ for r = 2 t o n - l ]
and
a'
= a[1 +
[for r = 1 and r = n ]
(4.2.74)
91
Calculate the scaling factor at each node in the
filter from
n
r
= [CL a/[tan(aw ) (Y
r
o r
+ Y
,
+ Y
r-l,r
r,r+l
(4.2.75)
Calculate the coupling admittance between each stub
from
Y
,
r ,r+l
r r+l ---- r
= -----K r,r+l
n rn r+l
n at°o
(4.2.76)
Calculate the admittance to ground of each stub from
y
-Yb tan (bu )
= — ------- °
r
tan(awQ )
_ y
r-l,r
y
r,r+l
(4.2.77)
Finally the admittances of the transformer elements
(of electrical length aw)
Y
o
= 1 -
are calculated from
l /nnsin(aw )
1
o
(4.2.78]
Y^, =
01
1/n, sin (aw )
1
o
(4.2.79)
Y ' =
1
1 / n 2 - l / n 1sin(aw )
1
1
o
(4.2.80)
4.3
Design of Tunable Bandpass Filters
4.3.1
Computer Analysis of Tunable Bandpass Filters
In order to predict the performance of varactor tuned
and fixed frequency filters computer analysis programs were
92
developed.
S.S.S.
The varactor equivalent circuit model and the
losses were included in the programs as described
in chapter three.
Figures 4.3.1 - 4.3.3 show the computed midband
dissipation loss of varactor tuned combline filters as a
function of the passband bandwidth,
and bias voltage.
the degree of the filter,
The results were obtained from the
analysis of filters with a centre frequency of 4 G H z , a
passband return loss of 15dB and with the MA46620G varactor
as the tuning element.
The S.S.S loss was not included in
this analysis as it is primarily caused by lossy short
circuits which are not ammenable to analysis.
In order to use the data presented in Figs.
4.3.1
to 4.3.3 to predict the insertion loss of a filter with a
midband frequency other than 4GHz the bandwidth of the filter
must be scaled by a factor 4/fo where fo is the centre
frequency of the filter in GHz.
Computer analysis of the tunable open circuited
combline filter is not presented.
This is because in
Appen di x 4.2 it is shown that the Q factor of this filter
is approximately double that of the tunable combline filter.
Thus,
the loss of this filter will be the same as that for
a combline filter with double the bandwidth.
The computed midband dissipation loss of the distributed
combline filter is shown in Fig.
in Fig.
4.3.4.
The data presented
4.3.4 is for 4GHz filters with 0.3" ground plane
spacings.
The analysis of filters with other centre frequencies
and ground plane spacings is easily accomplished since the
Fig.
4.3.1
Bias
4V
Bias
30V
Bandwidth
Bias
0V
0.5%
19. 7dB
14. 7dB
9.98dB
1%
11.9dB
8 .4dB
5. 5dB
2%
6 .5dB
4. 4dB
3. ldB
5%
2.65dB
1.8dB
1.5dB
10%
1. 4dB
0.97dB
0.7dB
Computed Mid-Band Loss of Varactor tuned
Combline Filters of degree two as a function
of passband bandwidth and bias voltage
Bias
OV
Bias
4V
Bias
30V
0.5%
37.5dB
29dB
23.1dB
1%
23.0dB
17.2dB
12.9dB
2%
13.4dB
5%
9.3dB
6.1dB
4. 9dB
5.7
3.7dB
3. ldB
Bandwidtl
10%
Fig.
4.3.2
9.2
7.1
Co mpu ted Mid-Band Loss of Varactor tuned
Combline Filters of degree three
Fig.
4.3.3
Bandwidth
Bias
OV
Bias
4V
Bias
30V
0.5
55
44
34
1
36.2
26.8
20
2
20.9
14.5
10.8
5
9
5.9
4.4
10
4.65
3.2
2.5
Computed Mid-Band Loss of Varactor tuned
Combline Filters of degree 4
Bandwidth
Fig.
4.3.4
Degree
3
Degree
5
Degree
7
Degree
9
Degree
11
0.5%
1.8dB
11 dB
19.9dB
33.6dB
49.7dB
1%
0.6dB
3. 5dB
7.7dB
14.9dB
23.3dB
2%
0.26dB
1.66dB
3. 85dB
6.9dB
ll.ldB
5%
0.17dB
0.6dB
1.42dB
2.59dB
4. ldB
10%
0.09dB
0.3dB
0.65dB
1.18dB
1.86dB
Computed Mid-Band Loss of 4 GHz distributed
combline filters as a function of bandwidth and
degree.
These results are for a ground plane
spacing of 0.3"
93
filter Q factor is proportional to the ground plane spacing
and proportional to the square root of frequency.
4 -3.2. Design of a Varactor Tuned Combline Filter
The initial aim in varactor tuned bandpass filter
design was to achieve octave tuning.
This requires a 6:1
capacitance ratio to tune from 0 = 3 0 ° to 0 =60°.
Varactors
with chip capacitance ratios of 8:1 are available but these
capacitance ratios are severely degraded when the chip is
enclosed in a microwave package.
For example,
varactor has Cj (o) = 0.8pF and Cj (-60)
= O.lpF.
the MA 46620G
This varactor
is enclosed in a stripline package with a capacitance of 0.18pF.
The terminal capacitance ratio of the packaged varactor is
thus reduced to 3.5:1.
This capacitance ratio will provide
tuning from 0 = 35° to 0 = 55° which is considerably less than
an octave range.
Attempts to remove package effects by
bonding varactor chips directly onto the S.S.S circuit board
have proved unsuccessful because of poor contacts and the
long highly inductive bandwire required.
Octave tuning would
thus seem to be restricted to lower frequencies where higher
capacitance varactors can be used.
For example, hyperabrupt
varactors with zero bias capacitances of 4pF and chip capacitance
ratios of 10:1 are commercially available.
Even when packaged,
these devices will exhibit terminal capacitance ratios of
7:1.
Unfortunately,
the Q factors of these varactors are
too low for the application of broadband tuning of microwave
filters above frequencies of 1 G H z .
94
The design objective was thus to realise a tunable
combline filter capable of tuning from 6 = 35° to 9 = 55°.
The centre frequency of this filter was to be 4GHz and the
mid ba nd dissipation loss was to be less than 6dB.
The loss
specification which seems somewhat arbitrary, was chosen
because it is similar to specifications offered by
ma nufacturers of U.H.F.
Fig.
varactor tuned filters.
Examining
4.3.1 we see that a two resonator filter with a 5%
passband b andwidth tunable about 4GHz, will have a maximum
m i d b an d loss of 3dB.
This is the computed loss contributed
by the varactors.
Further loss will be contributed by the
short circuits of
the S.S.S resonators and by the varactor
contact resistance.
The total extra loss will probably be
less than 2dB, thus
this specification will satisfy the loss
requirement of 6dB maximum.
The design process in section 4.2.1 was used to design
the filter to the following specifications
Centre frequency
4GHz
Passband bandwidth
200 MHz
Passband return loss
15dB
Varactor capacitance
0. 6pF
Length of resonators
at mid-band
45°
Nu mb er of resonators
2
The varactor capacitance used in the design process
was 10% less than the geometric mean of the zero bias and 60V
varactor c a p a c i t a n c e s .
This value was used to compensate
for the var actor bondwire inductance which tends to tune the
filter down in frequency.
95
The admittances of the stubs in the combline filter
were calculated to be as follows
YQ
=
0.6607
Y Q1
=
0.3393
Y1 = Y2
=
0.4517
Y 12
=
0.0668
(for a 1 ft s y s t e m ) :
Calculations of the Dimensions of the Combline Filter
The procedure for calculating the physical dimensions
of the combline filter is somewhat different to previously
presented m ethods
because the lines are coupled,
this
procedure is now presented in detail.
Firstly,
unit length)
the static capacitances of the stubs
(per
are calculated using
C/e
=
(4.3.1)
7.54 Y
Thus
4.982
c o/e
II
u
\
iH
o
u
C 1/e = C 2/e
C l2 /e
=
2. 558
3.4058
0.5037
Next the normalised coupling gaps between the stubs
can be obtained by using the values of C ^
graph of AC/e as
values are
and C ^2 in the
a function of s/b in Appendix 2.1.
These
96
=
0.015
A ground plane spacing of 0.3" was to be used,
SOI
=
0.0045"
S12
=
0.0915"
Next,
thus
the widths of the bars must be evaluated.
The
capacitance to ground of an S.S.S bar is affected by its
proximity to n eig hbouring bars.
The total capacitance per
unit length to ground of such a bar is expressed by
|
=
+ 2Cf'ei + 2 C f ’e 2
4*
(4.3.2)
where Cf'e^ and C f ' e 2 are the fringing capacitances from
one corner and half the associated vertical side wall of an
S.S.S
bar to ground when an open circuit is placed along the
line midway between a bar and its nearbouring bar.
These
fringing capacitances are known as the even mode fringing
capacitances.
The even mode fringing capacitances are
presented as a function of the separation between the bars
in Appendix 2.1.
For the transformer stubs at the input and output of
the filter we have
Cf'e^ and Cf'
From Appendix 2.1
Cf'e2
=
0.014
=
0.45
97
Rearranging 4.3.2 we obtain
w
|[| - 2Cf'e 1 - 2Cf' e2 ]
(4.3.3)
Thus for the trans for mer stubs
w
w
^^[4.982
o
-0.9 -0.028]
0.304"
o
For the resonators the even mode fringing capacitances
are
Cf'e1
=
0.014
Cf'e2
=
0.23
Thus,
from 4.3.2
W 1 = w 2 = 0.218"
The calculation of the lengths of the resonators was
relatively simple.
These were designed to be 45° long at
4GHz which is a length of 0.369".
The lengths were then
modified to account for fringing from the ends using equation
2.1.13 as follows
L + L - 0 . 23b
With b = 0.3" we obtain
L = 0.3"
(4.3.4)
98
Design of bias chokes
The combline resonators are short circuited at one
end by clamping them into the side walls of the S.S.S
housing.
the
Thus a d.c.
resonator.
short circuit exists at one end of
Consequently,
the r.f short circuit needed
by the varactors cannot be realised by a d.c.
short circuit,
otherwise both terminals of the varactor would be short
circuited to d.c.
The r.f.
short circuit to the varactors
was provided by connecting a lOpF chip capacitor in series
with the varactors to ground.
The bias filter was then
connected at the junction of the chip capacitor and the
varactor.
The bias filter design was otherwise similar to
that described in
circuit board
section 4.3.2.
The layout of the printed
of the filter is shown in Fig.
4.3.6.
The design of the S.S.S housing was essentially similar
to that for the bandstop filters.
The housing was gold plated
to provide good ohmic connections for the short circuits.
The assembled filter with the top half removed is shown in
photograph no.
3.
Measured Performance of the Varactor Tuned Combline
Filter:
The frequency response of this filter is shown in
Figs.
4.3.7 - 4.3.9 for bias voltages of OV, 4V and 30V.
The performance of this device is summarised below
Bias Voltages = OV, OV
Pass ba nd
Centre Frequency
3.2 GHz
Passband Bandwidth
186 MHz
Passband Insertion Loss
5.4 dB
Passband Return Loss
8 . 1 dB
Stopband
Fo + 400 MHz
2 4 dB
Fo + 1000 MHz
3 8 dB
Bias Voltages = 4V,
3.8V
Passband
Centre Frequency
4.16 GHz
Passband Bandwidth
209 MHz
Passband Insertion Loss
4. 1 dB
Passband Return Loss
10.5 dB
Stopband
Fo + 400 MHz
26 dB
Fo + 1000 MHz
39 dB
Bias Voltages = 30V,
25.6V
Passband
Centre Frequency
4 .8 GHz
Passband Bandwidth
189 MHz
Passband Insertion Loss
4.2 dB
Passband Return Loss
9.1 dB
lOO
Stopband
Fo + 400 MHz
26 dB
Fo + 1000 MHz
37 dB
The tuning bandwidth of the combline filter was slightly
less than expected.
This was a result of the fringing
capacitance from the ends of the combline resonators to
ground.
This capacitance appears in parallel with the
varactor thus effectively increasing the parasitic package
capacitance.
The value of the fringing capacitance was
calculated to be 0.045 pF which is 20% of the varactor
package capacitance.
Experiments involving rounding the
corners of the combline resonators to minimise this capacitance
have proved unsuccessful and it woul d thus appear that the
fringing must be accepted.
Large Signal Effects
Second harmon ic distortion was measured using the
same techniques as in section 3.5.
cept point was + 7
volts bias.
The 2nd ordered inter­
dBm at 1 volt bias and Hh31 dBm at 10
This is approximately 6 dBm lower than that
of the bands to p filter and is explained by the fact that
the varactors
are more strongly coupled into the bandpass
filter than the bandstop filter.
distortion was observed.
Again, no third harmonic
Fig.
4.3.5
Realisation of bias filter
0.05
Fig.
4.3.6
The dimensions of the Printed Circuit
of the varactor tuned Combline Filter
Photograph no.
3
The V aractor tuned Combline Filter
03
s i s's
Fig,
4.3.7
-300MHz
Frequency
-20CMHz
Response
-lOOMHz
of
Varactor
3.2 GHz
tuned
Filter
+20CMHZ
Combline
+10CMHZ
at
zero
+300MHz
bids
aiss
cn
Fig.
4.3.8
Frequency
Response
of
varactor
tuned
Combline
Filter
with
approximately
4V
bias
cn
to
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101
4.3.3
Design of a Varactor Tuned Open Circuited Combline
Filter
A varacto r tuned open circuited combline filter was
designed and constructed to the following specifications.
Centre frequency
4GHz
Tuning bandwidth
1GHz
Passband bandw id th
120MHz
Passb an d return loss
15dB
Number of resonators
3
Pa ssband insertion loss
< 7dB
The computed midband insertion loss of this filter
was less than 5dB
at any tuned frequency.
The element values
of the combline array were evaluated using the technique
described in section 4.2.2.
The element values were
Yq
=
0.683
Y q1
=
0.3108
Y1 = Y3
=
0.3328
Y 12
=
0.0494
Y2
=
0.5518
The physical design and construction of this filter
was essentially the
4.3.1.
same as the method described in section
The layout of the S.S.S circuit board of this filter
is shown in Fig. 4.3.loand a picture of the assembled device
is shown in photograph no.
4.
The me as ure d performance of this filter is summarised
below.
102
Bias Voltages = OV, 0.01V,
OV
Passband
Centre Frequency
31.51 GHz
Passband Bandwidth
108 MHz
Pa ss ban d Insertion Loss
6.1 dB
Passband Return Loss
9.1 dB
Stopband
Fo + 200 MHz
35 dB
Fo + 500 MHz
57 dB
Bias Voltages = 4V, 4.61V,
4.2V
Passband
Centre Frequency
3.9 GHz
Passband Bandwidth
118 MHz
Passband Insertion Loss
5.2 dB
Passband Return Loss
11 dB
Stopband
Fo + 200 MHz
3 7 dB
Fo + 500 MHz
59 dB
Bias Voltages
30 V, 38.7V,
32V
Passband
Centre Frequency
4.43 GHz
Passband Bandwidth
112 MHz
Passband Insertion Loss
4.3 dB
Passband Return Loss
10.1 dB
Stopband
Fo + 200 MHz
36 dB
Fo + 500 MHz
54 dB
1.394"
i
0.15"
0
4
“tl
1143
(- -)
D. 228"
-»l
<- -3
r0.117
1.03"
2.018"
b.05"
0.1
Fig.
4.3.10
0.05"
Dimensions of the circuit board of the open
circuited combline varactor tuned filter
Photograph no.
4
The V aractor tuned Open Circuited Combline Filter
103
4.3.4
Design of Fi xed Frequency Di stributed Combline Filters
Two distributed combline filters were designed to the
following specifications
Design 1
Centre frequency
4 GHz
Passband bandwidth
50 MHz
Passband return loss
15 dB
Number of resonators
3
Design 2
Centre frequency
4 GHz
Passband bandwidth
200 MHz
Passband return loss
15 dB
Number of resonators
3
Design 1
The element values for the distributed combline filter
were computed using the procedure in section 4.2.3.
admittances of the long resonators,
136° w er e chosen to be 0.5
Yq
=
0.746
Y 0l
=
0.232
=
0.572
Y 12 — Y 23
=
0.026
Y2
=
0.781
=
Y3
of electrical length
The admittances of the array
of stubs of length 36° were
Y1
The
104
A ground plane spacing of 0.3" was chosen and the
physical dimensions of the filter were calculated using the
same procedure as for the tunable combline filter.
Correction for Coupling Between Resonators
In the design process,
the 136° long stubs were
assumed to be isolated from each other.
In practice this
is not the case and they were 0.387" apart.
to a coupling admittance
This corresponds
(normalised to 1ft) of 0.0027
The unwanted coupling will alter the response of the filter.
The coupling can be removed by inserting walls in the S.S.S
housing between the stubs.
This is, of course,
a complicated
mechanical procedure and it is simpler to alter the coupling
between the 36° long stubs to compensate for the unwanted
coupling.
The coupling admittance between the 36° long stubs
was 0.02 6
and thus the total coupling admittance between
the resonators
Y
at 4GHz was
=
0.025 tan ( 36)
=
0.0188 - 0.0026
+
0.0027 tan (136)
(4.3.5)
The admittance between the 36° stubs should thus be increased
by 0.0026 at 4GHz.
Thus Y 12t a n (36)
or
Y 2
=
=
0.0214
0.295
The m odified circuit board dimensions of this filter are
shown in Fig. 4.3.11. A picture of the assembled filter is shown
in photograph no.
5.
1.912"
a
0.246"
0.35"
T
0.265"
0.338"
5—
—3
t —
Jbl62’lt,
-
0 .01 "
0.147"
0.35"
E—
— ^
0.375 "
1.23"
Fig.
4.3.11
Dimensions of the Narrow Band Distributed
Combline Filter
t
ws
Fig.
4.3.12
Broadside Coupled Strips used for realising
high values of coupling admittance
Photograph no.
5
The 4GHz D i st rib ut ed Combline Filter
105
Design 2
The element values of this filter were calculated
using the procedure described in section 4.2.3.
The lengths
of the short resonators were 20° at 4GHz and the long
resonators were 13 7° long.
The long resonators were chosen
to have a characteristic admittance of Yb = 0.45 and the
admittances of the remaining elements
system)
(normalised to a 10
are shown below.
Y
o
Yoi
Y
= 0.273
“ ° v 72
= Y 3 = 0. 493
YI 2 = Y2 3 = ° ’167
Y,
= 0.991
The design of the physical dimensions of the S.S.S
circuit bo ar d wer e similar to that for design 1 wit h one
exception.
The value of Y ^ of 0.72 was relatively high
and could not be realised by side coupled strips.
This
coupling was achieved by overlapping strips on opposite sides
of the dielectric circuit board [ F i g .4.3.12]. An approximate
design procedure for calculating the overlap w
s
coupling capacitance AC is presented in ref 4.4.
for a given
This
procedure assumes zero thickness conductors and a homogeneous
dielectric medium.
In practice,
the coupled lines operate
in an inhomogeneous dielectric medium, but since the dominant
coupling fields occur mainly in the dielectric circuit board,
this will not be a serious problem.
is calculated as follows
The overlap distance w g
106
=
1
[| loge(|)
+
(l-t/b) loge
(^|) ]
(4.3.6)
where
q = K/a
(4.3.7)
K = [exp(!|^) - 1]
(4.3.8)
a =
/
t / b - K .2
(t 7 b T T ) + K
,t/b-K
t7b+T
,
(4.3.9)
In these equations AC is the required coupling capacitance
defined by
7 54 Y
AC
=
0.1
/ e
However,
(4.3.10)
r
the effective electrical length of a transmiss­
ion line in a dielectric is increased by a factor >/re~f thus in
order to preserve the correct coupling admittance at the m i d ­
band frequency of the filter AC must be modified as below
AC
=
7.54 Y
-------—
er
(4.3.11)
Note that 4.3.11 is only valid for short electrical lengths
of transmission line
(0 <45°) when the tangential frequency
behaviour is not significant.
In this particular example
this condition is satisfied.
For Y q1 = 0.72 then
AC
=
2.467
Using this value of AC and applying 4.3.6 - 4.3.10 we obtain
ws = -0.027"
Where the negative sign indicates that the lines are
offset rather than overlapping.
The remainder of the design of this filter was similar
to that for design 1.
board are shown in Fig.
The final dimensions of the circuit
4.3.13.
0.243"
0.535"
0.062"
<—
— » < ------ >
-- >
Fig.
4.3.13
<—
Dimensions of the 5% Bandwidth Distributed
Combline Filter
107
Me as u r e d Performances of the Fixed Frequency Bandpass
Filters
Design 1
Initially,
response.
this filter exhibited an undercoupled
This was caused by fringing capacitance between
the ends of the tr ansformer elements and the first and last
stubs which increased the effective value of Y q ^ .
To
compensate for this fringing the coupling gaps at the input
and output of the filter were increased until an equiripple
response was observed.
were 0 .0 2 1 ", i.e.
This occurred when the coupling gaps
approximately double their designed value.
The response of this filter is shown in Fig.
4.3. (4- and is
summarised below.
Passband
Centre frequency
4 GHz
Bandwidth
4 5MHz
Return loss
15 dB
Insertion loss
2 dB
Stopband
Fo + lOOMHz
35 dB
Fo + 200 MHz
50 dB
Design 2
This filter
perf or med according to the specification
at the first attempt.
is summarised below
The response,
shown in Fig.
4.3.15
of
1%
N
N
I
LO
N
’ln
ri
in
in
slal
CO
O
CN
Bandwidth
LT>
the
CM
Response
Distributed
ln
Frequency
N
4.3.14
m
r+
Fig.
Filter
LO
a
+
CM
I
T3
< c os m
i4 us j a?
m
Pig.
4.3.15
Frequency
Response
of
the
5%
Bandwidth
Distributed
Filter
108
Passband
Centre
frequency
4 GHz
Bandwidth
200MHz
Return loss
17 dB
Insertion loss
0.4 dB
Stopband
4.4
Fo
+
400 MHz
2 8 dB
Fo
+
800 MHz
4 5 dB
Conclusions
A novel design theory has been developed for tunable
combline filters capable of octave tuning with approximately
constant response shape.
presented.
Explicit design formulae are
Com puter analysis of varactor tuned combline
filters is presented enabling the performance of any device
to be predicted.
Experimental devices have been constructed
and the results exhibit close agreement with theory.
The basic tunable combline filter theory is extended
to the design of open circuited combline and distributed
combline filters.
Computer analysis and experimental p e r ­
formances of these devices are also presented.
109
References
4.1
"Theory of Mic r o w a v e Coupled Line Networks"
J.O.
Scanlan
I.E.E.E.
4.2
Proceedings Vol 68 No 2 FEB 1980
'Combline Filters of Narrow or Moderate Bandwidth'
G.L. Matthaei
Microwave Journal, Vol 6 pp82-91
4.3
(AUG 1963)
'Theory of Electrical Filters'
pp 48-49
J.D.
Rhodes
Wiley
4.4
'Impedance of Offset Parallel Coupled Strip Transmission
Lines'
J.P.
Shelton Jr.
I.E.E.E.
Trans MTT VOL MTT-14 No 1 JAN 196 6
110
Appendix 4.1
Derivation of the values of the transformer elements
in the combline filter.
The input/output coupling network is shown in Fig.
4.2.5. It is
assumed,
symmetrical,
i.e. Y Q = Y ' ^ .
of this network,
for simplicity,
that the network is
Evaluating the input admittance
looking back into the load via Y
, and
after scaling by the factor tan(ani)/tan(aaio ) we obtain the
real and imaginary parts below.
ReYin(jw)
tan (aw)
=
(A.4.1)
tan(aw )[(1+Y /Ynn)2 + tan2 (aw)/Y 2 ]
o
o 01
ul
|I m Yin (j to)
-[ <l+YoA 01> Yo (2 +
+ tan2 (aw) (1 +
tan (au)Q )[ (1 + Y0 /Y0 ^
2
(A.4.2)
+
For a per fe ct match at
w
=
w q
we require
(A.4.3)
1
iReYin (iw )
1
o
o
Im Yin (j uo )
(A.4.4)
From A . 4.3, A . 4.4 and A . 4.1 we obtain
11 + V
or
Y
Y 0 1 >2 + t a n 2 (a»o )/Y 0l 2 - 1
2
0
0
+
2
Y
0
Y 0 1
+
t
a
n
2
( “
0
)
(A.4.5)
(A. 4. 6 )
Ill
Substituting A . 4 .6 into A . 4.1 we obtain
tan (aw)
Re Yin (j w)
=
tan (aw )[1 -
tan2 (aw)
+ tan
01
(A.4.7)
(aw )-|
o J
loi
With aw = 6 and after some manipulation we obtain
jFfeYin (jw) =
Y
2 sin(6)cos(6)cos ^ (
---------- ^-------------------- T ------- 2----sin (0 )[Y-,, cos (0)cos (0 ) + cos (0 ) - cos (0)]
o
01
o
o
(A. 4. 8 )
Examining A . 4 .8 we see that if the denominator were made
frequency invarient we wo u l d obtain the required slow
frequency variation of s i n (0)c o s (0) = sin(20)/2.
To remove
the frequency dependence of the denominator we have
Y q1
=
1/cos (0q )
(A.4.9)
Thus
|ReYin(3 “ )
*
sin (2 6 ’ )
o
(A.4 .10)
Substituting A . 4.9 into A . 4 .6 we obtain
YQ
=
1 - 1 / c o s ( 6q )
(A.4.11)
112
Appendix 4.2
A Comparison of the Q Factors of Varactor Tuned Combline and
Open Circuited Combline Filters
The Q factor of a shunt resonator composed of a
conductance G in parallel with a susceptance B(o)) can be
expressed as follows.
0)
Q
where
oj
is
o
2G
=
(B (to) )
0) = 03,
(A.4.12)
the resonant frequency of the circuit.
The basic resonators of the combline and open circuited
combline filters are shown in Figs. A . 4.1 and A . 4.2.
In
order to evaluate the Q factors of these resonators it is
necessary to have a shunt model for the varactor.
The admittance of the varactor is
Y
=
R -j/o)C
+ j3
(A.4.13)
=
G
(A.4.14)
G
=
- ^ ^ )2 ■
(1 + (RoiC)
(A.4.15)
x
_
joiC______
(A.4.16)
where
(1 +
Now
R ojC
=
1/Q
(RojC) 2 )
(A. 4.17)
7 T C
S/C STUB
_
c
Fig. A . 4.1
The Com bline Varactor Tuned Resonator
Fig. A . 4.2
The Open Circuited Combline Varactor Tuned
Resonator
113
where Q is the varactor Q factor.
Normally Q >> 1
G - R (oiC)2
Thus
(A.4 .18)
jB = juC
(A.4 .19)
Using the shunt varactor model in the combline filter we
obtain
B(oi) = joiC - jY/tan(aw)
(A.4 .2 0 )
Thus
i f <"L =
o(jl)
Now
Y
=
Letting Gc =
c
+
^/ \
s m (aw)
u) Ctan(ao) )
o
o
(A. 4 .2 2 )
ao) , from A. 4. 21 and A. 4.22 we obtain
*9|al)“i|01=01o = C[1 + sc-4
rfnc-NJ
m (2 0 )
Similarly,
(A. 4 .21 )
(A-4.23)
for the open circuited combline filter, with
0oc = & to., we obtain
9B (pi)
3oi
co = o)
Thus,
=
C [ 1 --- --------r ]
sin (20 oc)
(A.4 .24)
for similar varactors, the ratio of the Q factors of
the two filters is
given by
114
[1 + -- 26_£--- -j
sin(2©c)
=
Qoc
r, _
29oc____
sin (26oc)
In particular for ec = 45° and 0oc = 135° we obtain from
A . 4.25 that
Qoc
=
2.22Qc
- 115 -
5. GENERAL CONCLUSIONS
This thesis has been devoted to a study of the design and
realisation of varactor tuned microwave filters.
In partic­
ular, both bandpass and bandstop filters, with narrow pass­
band and stopband bandwidths and broad tuning bandwidths are
investigated.
In chapter one the advantages of varactor diodes as compared
to YIG devices for the tuning of microwave filters are dis­
cussed.
A brief introduction to varactor physics and an exam­
ination of the specifications of commercially available varactor
tuned filters shows that with the use of GaAs varactors,
the
operating frequencies of varactor tuned filters can be consider­
ably extended.
Also in chapter one, various realisation techniques for microwave
filters are discussed and it is concluded that varactor tuned
filters are best suited to realisation using Suspended Substrate
Stripline techniques.
In chapter two initial experimental varactor tuned filters are
discussed.
Methods of characterisation of varactor diodes are
presented,thus enabling correct choice of a varactor for a given
requirement.
Initial work concentrated on the relatively simple problem of
tuning bandstop resonators.
A novel bandstop resonator realised
in Suspended Substrate Stripline is presented.
Experimental
characterisation of this resonator is presented thus enabling
exact design to be performed.
Several varactor tuned resonators
have been constructed and the experimental performances of these
devices show the various trade offs between insertion loss, stop­
band bandwidth and tuning bandwidth.
The design of varactor bias
filters is also discussed in this section.
-
116 -
Next in chapter two, the design of varactor tuned bandstop filters
is discussed.
These filters were composed of the basic bandstop
resonators separated at intervals along a uniform impedance main
line.
These Filters exhibited a frequency response which was
symmetrical only at a single tuned frequency^ and was skewed at
frequencies above and below that frequencyt Network Analyser measure­
ments showed that the phase angle between the resonators of the
filter at that particular frequency was not 90^ as would normally be
expected.
This was a consequence of the fact that each single reson­
ator possessed a skewed frequency response and a phase shift between
the resonators not equal to 90^ was required to compensate for the
individual resonators and thus produce a symmetrical filter response.
It thus became obvious that a design technique for calculation of the
correct phase shift between the bandstop resonators was required.
Finally in chapter two, the design of varactor tuned bandpass filters
is discussed.
In particular the experimental performance of a two
cavity filter is presented.
This filter exhibited an equiripple
response at a single frequency, however, tuning the filter only 5%
away from that frequency produced a drastic deterioration in pass­
band response-.
This phenomenon was a result of the tangential
frequency variation of the coupling between the filters resonators.
Thus, if broadband tuning were to be achieved, a technique for
externally compensating for the distributed coupling between the
resonators would be required.
In chapter three the design and physical realisation of varactor
tuned bandstop filters is described.
Firstly, a novel technique
for evaluating the correct phase shift between bandstop resonators
in order to achieve the optimum symmetrical response is presented.
This method results in simple recurrance formulae, which are easy
to evaluate using a calculator.
The design and experimental per­
formance of two fixed frequency bandstop filters designed using this
technique is presented.
Computer loss analysis of these filters is
presented, enabling the simple evaluation of the selectivity of these
filters.
Next in chapter three the design of a three cavity varactor
tuned bandstop filter tunable from 3.5 GHz to 4.5 GHz is presented.
This device performed in very close agreement with computer predictions.
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In Chapter four, the design of varactor tuned bandpass filters is
presented.
Firstly a novel technique for the design of combline
filters is presented.
This technique uses input and output
coupling mechanisms to the filter which compensate for the
frequency variation of the coupling between its resonators,
enabling broadband
thus
tuning to be achieved with minimum degrada­
tion in passband performance.
In addition this filter possesses
the important property of retaining almost constant passband band­
width across very broad tuning bandwidths.
A n experimental two
cavity device of this type achieved 1600 MHz tuning around 4 GHz
with a constant return loss and almost constant passband b and ­
width across the entire tuning bandwidth.
Computer analysis of
this varactor tuned bandpass filter is presented enabling deter­
mination of the passband loss as a function of centre frequency,
degree and passband bandwidth.
The design theory of tunable combline filters is also extended to
include two other types of filter in chapter four.
These are a
tunable combline filter with open circuited resonators and a
distributed fixed frequency combline.
The former type of device
possesses a higher Q factor than the normal combline filter allbeit
with a loss of tuning bandwidth.
The distributed combline filter design
is a useful technique for realising a narrow band fixed frequency
bandpass response.
This type of filter is relatively insensitive to
manufacturing tolerances.
Several experimental fixed frequency and
varactor tuned open circuited combline filters have been constructed
and their performances are presented along with computer predictions
of passband insertion loss.
In summary, the design and construction techniques for varactor tuned
microwave bandpass and bandstop filters have been presented.
Experimen­
tal devices have been constructed and these exhibit low loss and broad
tuning bandwidths.
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118
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FUTURE WORK
In order to achieve octave tuning of microwave varactor tuned
filters a terminal varactor capacitance ratio of 6:1 is required.
This involves removing the package capacitance from the varactors.
To achieve this end chip varactors have been examined.
The
problem with these devices is a large increase in parasitic induc­
tance due to the long bondwire required to connect the top of the
chip to the duroid.
A better alternative for future work would
be to use beam lead varactors which could be manufactured with
very low package capacitance and inductance.